For used GPUs RX480(80$ish) or R9 Nano(100$) are your friends, alternatively for Nvidea the 970(80-90ish$) will be your friend, if you had 200+ to spend on used GPU and a good PSU then a used Vega are apparently really fun to tweak, usually undervolting core and overclock the HBM VRAM being the beez knees with any of AMD GPUs with older power hungry architecture, but gifted with HBM VRAM. If that sorta tweaking sounds fun but 200$ is too much then that is why i mentioned R9 Nano. Problem with cards that use HBM(2) is they were often very popular for mining, which by itself isn't the worst, but likely means they could come with vbios setup for mining and even if you can install correct vbious sometimes they'll just konk out after the fact. With any of these cards get atleast a decent 80+ Bronze 450W power supply, usually can find them 45$.
Can you overclock with motherboard that came with thinkcentre? I wouldn't imagine not with most OEM sourced boards, not sure what board came with it though. If you want to use that motherboard and it can't overclock i'd highly recommend a 3770 over a 3570k, the extra threads help alot for reducing stuttering in modern games that often look for more then 4 threads.
Actually, i used to have a 3570 (none K, 3.8ghz iirc, paired with 1050ti i regret purchasing instead of rx570) and honestly i'd just go with a 3770 even if you can't overclock it because the stuttering can be a real problem, especially with games that use newer graphics engines (DX12 in particular.), no point in pushing high frames in a FPS if quickly looking around side to side causes huge, intermittent, frame-rate dips.