My basement is fully finshed but when I am using my PC or shop vac the breaker goes out on me what could cause this

If the power comes and goes, sounds like a loose connections usually at one of the receptacles.

Shop vacs draw 6-7 amps, almost 1/2 the capacity of a 15 amp breaker. If you have lights, TV, etc on the same circuit it is being overloaded. See if you can switch some outlets to another circuit. Do not put in a larger breaker!