I'm 6' and I have a older road bike I turned Single speed. I have $200 into it with $150 for the bike with great parts, and $40 into the single speed set up - 1/8 chain, crack screws, 16 tooth freewheel - And some grip tape. I chose this route because I'm taller and most single speeds are track type bikes. These bikes have a lower headstock and a higher seating position vs. the head stock - you don't want to be hunched over the entire time.
I HIGHLY RECOMMEND to find a older road bike that fits you, both seat and the headstock reach. Make sure it has horizontal drop outs, a clear body that the cables can be removed without playing with the body, make sure it has a tapered crank, and a good set of hubs and a good crank - you will grind these into the ground
Also, go for a single speed for your first one, then work on a fixie