I agree with bbattle. Building your own bike up from scratch is well worth it, and just for the learning experience alone. Sheldon Brown's website is a great reference for almost any question you may have.

Find a light good quality steel road bike. Replacing the rear wheel with a track wheelset won't set you back more than 150-200 $ if you are frugal. Then it is just a case of removing the front and rear derailleurs, the shifters and sorting out the chainline. One trick is to flip the outer chainring on to the inner side of the crank arm. That should give you a pretty good alignment. If you need more adjustment, go for a Phil Wood bottom bracket that has a bit of lateral adjustment, or a track crank.

Chain width doesn't matter too much. 1/8" looks cool.

All the rest can be replaced at will. Total cost should be around 450 - 500 $, satisfaction factor will be greater.