MAKING THE FILLER
Construction is as indicated on the drawing. (Craig used a tee fitting instead of a block of brass- a great idea! See below). Follow the instructions on the drawing above to ensure easy construction and that the flow paths are correct and unobstructed.   Some additional details:

The pre-tinning of the 1/4"  discharge tube noted in the drawing above is especially important since one can't apply solder to the joint in the typical manner due to the 3/8" tube that's already in place.   Details are provided below on how this is done.

If the potiential for lead in your brews worries you, de-lead the brass using the method John Palmer has posted about in the HB Digest. IMHO, it's not warranted for a filler since little brass is exposed to the beer and the contact time is relatively short.   Definitely use lead-free solder tho'.

The valves are not shown on the above drawing.  I once used cheap nylon pinch type valves on each of 3 vinyl filler tubes attached to the 1/4" tubes on the filler.  They are cheap but difficult to operate.  I now use ball valves on the co2-in and beer-in lines and a needle valve on the co2 vent line as per a tip  from Craig Sikes.  Don't waste time with clamp type valves.

The bottler uses 3/16" ID vinyl tubing for the CO2-in, CO2-out, and beer-in lines. (Hint for getting the tubing on the larger copper tubing: soak the copper and the last 3/8" or so of the vinyl tubing in very hot tap water. Remove from the water and grip the vinyl tubing with a hankerchief for a better grip and put it on the copper tubing with a twisting and pushing motion. I never use clamps and haven't had a blow-out yet..) The vinyl tubing on the business end of the filler reaches to just shy of the bottom of the bottles.

BOTTLE FILLING

SetUp
Get the brew in the keg ready by ensuring it's well carbonated and cold. For your first attempt, I recommend over-carbonating by a psig or two since you'll loose a tad of carbonation during the process. Prepare for bottling by sanitizing an abundance of bottles and the filler and it's ancilliary parts.  FWIW, I sanitize with Iodophor, drain briefly, invert and put them in a plastic box to drain a bit.   I seldom let all them dry completely.  I used to put the bottles in the fridge to cool before filling to reduce foaming but have stopped that PITA practice.   Make sure all of the valves are closed before proceeding. I once dispensed a couple of quarts of brew on the floor by failing to heed this advice... Rig up the bottler as shown in the drawing below.

You may have to play around a bit with the pressure regulator setting. The required setting depends mainly on the amount of friction loss in the beer-in line, the flow you use (those 2 are interrelated...) and the height of the brew in the keg above the bottle.

First a safety tip: always wear some sort of eye protection when bottling. I've never had a bottle bust, but, one never knows... I've read where others recommend wearing gloves. I don't since nimble teasing of co2-vent needle valve is required.

Bottle filling:

1. Put the filler into the bottle and hold it down firmly.
2. Purge purge oxygen from the bottle. Two methods: 1)  Open the co2-in valve enough to pressurize the bottle then close the valve, bleed-off the pressure and repeat several times or even  more if the brew is delicate, for compentitions, and/or you'll be storing it for awhile.  Pressurize a final time then shut the co2-in valve  to pressurize the bottle.  2) Leave both co2 valves open to purge, close the co2 vent valve then close the co2-in valve to pressurize the bottle.
3. With the bottle now at the same pressure as the keg, the beer-in valve is fully but slowly opened.  A bit of brew may initally flow into the bottle then cease.  Opening the beer-in valve fully reduces foaming as the bottle is being filled in the next step.
4. Tease open the co2-vent valve just a bit to bleed off pressure in the bottle.  This allows the brew to flow into the bottle slowly. If filling is too rapid, you'll get foaming which will reduce the carbonation level in the brew.  The goal is to have no foaming in the bottle during filling.  As a datum-point, it usually takes me around 10-15 secs. to fill a 12 oz. bottle.
5. When the bottle is about full (stop before the brew enters the 3/8" co2-vent tubing!),  close the co2-vent valve to stop flow then close the beer-in valve.  The order the valves are closed is important.
6. Slowy tweak open the co2-vent valve to depressurize the bottle. Opening to much/too quickly will cause foaming.  A tad more brew will flow into the bottle during this process and maybe out the vent line if you've overfilled the bottle.
7. Remove the filler. You'll note that the level in the bottle decreases as the filler is withdrawn (duh!)- that's why you want fill the bottle almost full. Cap the bottle ASAP.

Counter-pressure bottling may require some experimenting with the setup- espeically the co2 pressure and/or the elevation of the bottle in relation to the keg.  The best advice I can give is don't get discouraged- keep playing around/experimenting  until you get the hang of it.

One thing that reduces the loss of carbonation is to get the keg as cold as possible.  Submerging the keg in an ice-water bath is ideal but is a bit of a hassle so I don't do it.  One of the best ways to judge how much carbonation you lose in filling is to observe how much foam there is in the bottle while you are filling it.  At most,  I have maybe a 1/16" layer of foam atop the brew during filling.

Check Valves: You should have a check valve downstream of the CO2 regulator to protect it, however, additional check valves downstream in the CO2 lines can screw up filling- especially if you have a typcial spring-loaded check valve in the line to the filler and not one in the line to the keg- or vice-versa since the pressures will be out of whack.

To aid in observing for foaming and aid in determining when to stop filling dark colored bottles, put a light behind the bottle being filled.

Use something to hold the filler when you're setting up and when capping bottles.  This keeps it more sanitary and catches the inevitable drips.  I simply use a gallon jug filled with a bit of water for weight to avoid tipping.  Before use, it's also used for sanitizing the filler.

When you first start out, put a piece of vinyl tubing on the end of the co2-vent valve and route the free end to a slop container.  This ensures dribbles (or sprays if the valves are opened/closed in the wrong order!) don't mess up your bottling area.

Try not to enjoy too much homebrew while bottling- especially the first few times!

Going further  (capping on foam):

When I plan on storing the beer for a long time I cap on foam.  The goal is to complety fill the headspace of the bottle with foam.  This foam displaces all oxygen from the bottle and, if capped promptly/before the foam settles, ensures against oxidation of the brew during storage.  Here's how I do it:  After the bottle is filled, withdraw the filling tube so it's end is just under the surface of the brew at the top of the bottle.  Tease open the CO2-in valve.  This will push the beer remaining in the fill line out and foams the beer a bit.  The goal is to complety fill the headspace of the bottle with foam.  Cap promptly.  DO NOT attempt to completely fill the bottles with brew- if the bottles are subsequently warmed, they could leak for even break due to expansion of the brew in the absence of vapor space.  The downside of capping on the foam is it's a bit messier until you get the hang of it and don't over-flow bottles with foam.

Another way others suggest to cap on foam is to agitate or tap the bottle so some of the co2 comes out of solution and thereby  generates foam.   I don't like this method because one losses carbonation and, to me, it was very difficult to fill the head space without overflowing and making a mess.  BTW, I've read that the filling machines used by the big brewers generate foam for capping on by blasting the brew with water (or maybe brew?) via a a needle type affair inserted into the bottle.

Cleaning the filler
I just flush it with hot water via vinyl tubing/adapters.  Basically, I alternately connect it to each of the three lines and flush.  A bit o valve twiddling is required to ensure everything gets flushed.  I then disconnect the hot water line, open all of the valves, sling it and the connected tubing a bit to remove most of the flushing water and leave to air-dry.  If you use your filler alot, a periodic soak in cleaner (e.g. PBW) is a good idea.

Force carbonation and then counterpressure filling also works great for making your own pop. I'm partial to ginger ale. I use the recipe on the ginger ale extract bottle but add about 1 oz. of diced ginger to the hot extract/sugar/water mixture and steep at 170 degF for 30 minutes or so then pitch into enough cold water to make 4 gals. I force carbonate at 25-30 psig (depending on how big a hurry I'm in) and shake the keg then counter-pressure fill at 20 psig or so. Pop requires alot more carbonation than beer. Kids love carbonated kool-aid and it's alot cheaper than Cokes or other purchased sodas.
 

Basically, I just altered your plans a little. As suggested, ball valves except for the CO2 exit where I added a brass radiator drain cock as a bleed valve to the CO2 exit for better control - no "teasing" of the valve needed. Indeed, I almost don't have to open it, just allow its inefficiency to work for you. I fill slower than you, probably compensates well for the fact that I don't pre-chill the bottles.

For the brass block, I used a solid brass block "tee", drilling a 4th hole to make a "+". The left side is the "gas in", the right side is the "gas out". The beer flows in a straight line down through the top (I didn't want any angles to disturb the gas stability). I screwed in a male hose barb on the bottom, exit side -  a 7/32 brass rod fits in the inside of the hose barb perfectly (replacing the vinyl tube). I have several 7/32" rods of varying lengths for different sized bottles.

I thought I was going to have to grow another hand so I mounted the filler in a frame with the filler in a slide track so it lifts up and down, setting into the bottle leaving hands free to work the valves. The combination of the rubber stopper and the weight of the filler mount seems to be enough to keep the filler from being blown off - no clamp needed.

There is a bit of technique to the operation but after a few bottles (and beer baths) I can fill without mishap. Over charging the beer by a pound or two was a good tip. Excessive over charging is disasterous - gushers! For filling pressure, I pressurize to about 18 pounds - seems to work best at this pressure for me.

Bottling an entire 5 gallon keg finally convinced me to attach a handle to the filler body.  It's just a short piece of 1/4 brass rod soldered in a hole drilled in the top of the filler with a chuck of wood attached to the end of the rod.  It makes holding the filler down on the bottle much less tiring.   Still, if you plan on alot of bottle filling, Craig's sliding fixture is a great idea!  If you plan on bottling alot, maybe bench mount it, spring load or counter-weight the sliding filler or platform under the bottle and use rope/pulley or other linkage to connect to a foot treadle to move the botttle up into the filler or the filler down into the bottle. Could do the same with a bench capper while yo're at it...

I use a bit different filling process than given above.

• Purging the bottle before filling is easier (less valve twisting) and quicker using a new method: Open the co2 in valve to pressurize then crack the co2 out valve, let it purge for 10 seconds or so then close the co2 out valve and allow the bottle to pressurize before opening the beer line valve..
• When I plan on storing the beer for a long time I cap on foam:  After the bottle is filled, withdraw the filling tube so it's end is just under the surface of the brew at the top of the bottle.  Tease open the CO2-in valve.  This will push the beer remaining in the fill line out and foams the beer a bit.  Cap promptly.  This more nearly fills the bottle and, since the relatively smaller head space is filled with foam, there's  virtually no oxygen in the headspace.  DO NOT completely fill the bottles with brew- if the bottles are subsequently warmed, they could break due to the absence of head space.  The downside of capping on the foam is it's a bit messier until you get the hang of it and don't over-flow bottles with foam.
• If you don't use Craig's fixture for mounting the filler, use something to hold the filler when you're setting up and when capping bottles.  This keeps it more sanitary and catches the inevitable drips.  I simply use a gallon jug.  Before use, it's also used for sanitizing the filler.