If you want to build a car capable of F1 speeds yourself you can buy a used champcar crank up the boost of the engine (which gives you more than 1000 hp in a lightweight and small package), add some modifications to the aero you're there. Not particulary difficult to do with a budget sufficient to buy any regular, more expensive sportscar.
If we on the other hand suppose we should build the car from scratch with at least some concern about safety it will become more expensive naturally, but still nowhere near what even the back runners in F1 spend.
With a powerful engine we don't need particulary efficient aero, and without having to follow F1's aero regulations we should rather easily design a car that exceeds F1 standards in both downforce and aero efficiency.
To keep costs down the car should be built using off the shelves parts as far as possible; bespoke productions and custom orders will cost more due to the small series. For example; use a standard Hewland, Xtrac or similar 6 or 7 speed gearbox, not a CVT.
Covered wheels are prefered but not neccesary. A body separate from the chassi is probably a good idea (unlike F1), not because it is better but because if makes modifications to the car simpler. It would also be unwise to spend large sums of money on complex systems providing very little benefit; for instance avoid active ride, six wheels, four wheel drive. Keep it simple.
As for the engine I think there are many racing and production engines that are suitable. I do however think that very small displacements, very high engine speeds and very high boost pressures should be avoided. Also avoid engines with lots of torque as it will be difficult to find an off the shelves racing gearbox for such an engine. With an 'extreme' engine design you can end up spending a large portion of your budget solving engine related problems.
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