In general, first person shooters are some of the games that stress the GPU, MMO type games often stress the CPU more. You do need a CPU upgrade, however, a slightly faster video card will help as well, a 960 Ti for example. You can sell your current card on ebay after purchasing a faster video card.
If you only have 100 dollars to spend and want to upgrade your CPU, you will have to stay with Intel 3000 series CPUs, unless specified by your motherboard manufacturer. Go to your motherboard manufacturer site and upgrade your BIOS to the latest version, or a newer version that will suit your needs. The motherboard manufacturer web site should tell you what CPUs will work in your motherboard and BIOS version, upgrade your motherboard BIOS to at least the version necessary to run the CPU you want to purchase. You will need to know the exact motherboard type and version [often printed on one of the corners, but check your manual or motherboard web site to make sure of the location].
Look in your motherboard manual to find out how to enter BIOS [upon boot-up], your current version should be listed there, you can also download SiSoftware Sandra Lite [software] and it will tell you your BIOS version. [Click on the Hardware tab then Computer Overview - near the top left {green icon}, your BIOS will be one of the first items listed].
Download SiSoftware Sandra Lite here [it is free to use]:
http://www.majorgeeks.com/files/details/sisoftware_sandra_lite.html
Check websites for new and used CPUs. www.microcenter.com usually has good prices for CPUs, however, you often are required to go there to purchase it. If you go to ebay, make sure the seller has 5 or more sales and all 100% positive feedback. If the person has thousands of sales, 97% positive feedback is safe. The more % in positive feedback the better, it will be the right step in getting a good part.
I know this is a lot of information, however, I am trying primarily to keep you within budget of $100. Remember, you can sell your current CPU on ebay.com if you go over budget a little. If you do not sell your CPU for what you want out of it you used to be able to re-list it one more time without a fee. It has been a while since I sold something on ebay though.