I was looking to upgrade my graphics card, but I have a very limited budget of around $100. I found the AMD Radeon HD 7750 1GB GDDR5, which fits the $100 budget I have. After googling around for a bit I found that my processor would bottleneck the GPU, so I was wondering how much it would bottleneck it? Also if i upgraded to that card would I be able to run most modern games on 1280x720?
Try it, if the CPU turns out to be a limiting factor upgrade the whole package.
The problem is your budget which is very limiting. I'd say you should be able to play most games but you'd have some issues with games like BF3 Crysis 3 Farcry 3 even that they would still at least run on your system.
DO NOT UPGRADE! Your CPU will bottleneck the living heck out of any modern graphics card. My old dual core processor (E8500) was holding my 9800GT back. I didn't upgrade my graphics card until I had enough money to buy a quad core at the same time too. It's just a waste of money and time if you're going to just upgrade the graphics card now. Or if you have another family PC or a friend's PC that's running a quad core with the same socket, you can replace your CPU with theirs and then you can upgrade. Or buy a used quad core or something similar for a friendly price. Until then, if you don't have the money to upgrade the whole PC or at least the CPU don't think about upgrading your graphics card.