My personal suggestion is my current laptop. It should be around 600 dollars but it's really good. It's an ASUS X550JX-XX129D
Specs are:
Intel i5-4200h processor(2.8 to 3.4 GHZ with boost, 2 cpus, 4 threads, comes with Intel HD Graphics 4600)
1TB 7200RPM Hitachi HDD
4GB Onboard ram and you can add an extra 8gb in the free slot
A dedicated nVidia GT950M(freaking amazing. you find this GPU on laptops with prices 800+. amazing find on this laptop)
I can play most older games on max graphics for 60FPS. I play games like Dark Souls 3 on max, fallout 4 too(but not 60FPS, of course, around 40). Witcher 3's tough though, I can only play it at high with 30 at most. from 20-30 fps.
I never tried but I heard this PC can go GTA V max graphics at 60. I saw benchmarks with Black Ops 3 at max at 40 FPS(while recording, so it may get even better stats without that). There's youtube videos for this exact model tested on various games, go look it up!
For games like TF2 and CSGO, I bet it can deal with them without too much trouble. With that budget I believe this laptop will serve you well for quite some time.