Hmm,
if I had only 600 EUR/USD available, I'd only upgrade the SSD to a maybe faster and bigger Samsung Pro, m.2 maybe, so that game installs can live there. Stock up on RAM to atleast 16GB preferably 32GB and upgrade the graphics card. Given that the higher ranged cards cost easily 600 EUR, the budget is already used.
Bigger upgrades involve usually a new Mainboard, new CPU, new RAM, new cooler, new PSU. If you don't care about a non overclockable i5, ok, budget builds will be available
From my experience: my son uses my old trusty i7-4770K and achieves reasonable gaming performance with the mentioned SSD, RAM and GFX Upgrades.