I have a phenom 2 quad core and it is more than enough for my gtx 660 ti.
I wouldnt worry too much about bottlenecking the gpu, since your psu will be the most limiting factor in your decision.
If I had a $250 budget, I would go for a gtx 1060 6gb. It may bottleneck but since you are already on a budget it doesnt make sense to buy something worse around the same price. Just make sure that your psu is at least 400 w and has 6 pin connectors.