Get yourself a Lotus Elise 135R.
First race you should be able to do without any upgrades.
Otherwise, and if you afford it, buy all handling upgrades, gearing upgrades, race tires, aero packages and all that kinds of stuff but no engine upgrades/power upgrades.
For race 2: buy Powertrain Swap.
Race 3-10: additional upgrades within power restrictions.
Could use a Pagani Zonda and/or a Lamborghini Gallardo for last two races since they're containing a bit more powerful cars but they're still not as nimble as the Lotus, which with the right tune is like an extremely fast gocart.