I think you would better off buying a used shifter kart frame and a used 2 stroke dirt bike and combine them for those expectations. Problem with that kind of performance is the frame itself can't/won't hold up to it. If you build it strong enough from steel, then you will have to go bigger on the engine to overcome the weight, aluminum frame would do the job, but its expensive and hard to learn to weld aluminum.
I know a few racers who sold their 2 or 3 season old frames for next to dirt cheap without the engines, and would be more than enough for a street beater like you want, and the brakes and axles will already be able to take the beating you will be dishing out.
As far as an engine, the top speed is not the issue, the 0-60 time is. An 80-90cc 2 stroke will do 60+mph, but will take a bit to get there, a 125 will absolutely do what you want, except maybe the 5 seconds, a 250 will make you lose control of all bodily functions, resulting in mass cleanup efforts for you and anyone behind you.