Best cheap car...
Depending on what your idea of cheap is:
The RX7 (FD) is probably the best cheap car.
You can get one of many from the used car lot and compared to pretty much every car listed so far in this thread the RX7 (FD) is faster, handles better, and cost just about the same amount.
My nominations list will all come from Used Lot II (the newest of the used cars)...
RX7 (FD)
Skyline R34 '99
Supra RZ '97
Honda S2000
Those will all cost less than or right about 30,000 credits and all have the potential to pretty much walk on any of the road car races in GT4.
The Supra and Skyline are highlights though, since both will be capable of more than 800hp after serious modifications.
With that said, I'd like to take a moment to go over some of the Dealer purchased cars that would do well with this subject...
The Evo 8 MR and Impreza STi Spec C will both do a great job if you are prepared to tune a 4WD'er. The Corvette Z06 (C5) will also do wonders for you if there isn't a pre-existing bias against American cars. For that matter, the Z06 is probably one of the best cars in the game.
Even though the new cars are tempting, I suggest the budget racer sticks to the used lots. There are plenty of great used cars and if you are smart about your purchase there will be plenty of time and money for upgrades (since a good used Supra, Skyline, or RX7 will easily earn you more money and make you want to continue racing with that particular car).