How can you tell when one will bottleneck the other? <-- pls explain me

Recently my gpu went out so I'm looking to find a replacement that's either equal (or better if possible)

CPU: Intel Core i5 3570K
GPU: Msi Amd Radeon HD 7870
Motherboard: Msi Z77A-GD55
PSU XFX 550


Budget 150-300$

get an rx 580 4gb off fb or craigslist

Yeah something like an Rx 570/580 4gb would be in that price range and would be an improvement.
a 560 would be a little bit worse.