There are some good gaming laptops on the market but since a laptop with the same basic components as a desktop is already going to cost about 30% more, finding a good gaming laptop for anything even close to cheap is going to be a tough find. You may find something that is last years model or a floor display that is on clearance and heavily discounted. Or since it's the holidays there will probably be a few deep discount sales closer to xmas from retailers trying to clear out all their Xmas stock.
Basicly what your going to want to look for is going to be something that has these main basic elements, weather it's a desktop or a laptop.
An Intel i5 processor at least. i7 would be better but the performance improvements between i5 and i7 are not hugely noticeable especially if you get one of the faster Mhz i5 processors. i3 just really can't handle games well especially on a laptop.
anything running windows 10 64-bit your going to want AT LEAST 8GB of RAM, and 16 GB if you can. Especially on a laptop this will help things run more smoothly.
Some kind of separate video card that does NOT say anything about Intel Integrated Graphics <whatever> Intel Motherboard Integrated graphics is fine for web browsing and some web based games, but it bogs down ALOT on almost any Windows installed game. <Think Farmville 2 vs Sims 4 or Battlefield 1 and things like that> The currently popular laptop video card is the new Nvidia Geforce 1000 series cards for laptops, but also the Nvidia Mobility laptop cards which are a bit older should handle things pretty good. Basically if you see anything in the specs that talks about an NVidia grapics solution and NOT Intel Integrated your good.
Bear in mind looking around for a few mins the cheapest gaming laptop I could find was at Best Buy for $899..... so finding something for $600 as
@MrsFlynn said is going to be pretty tricky.
I wish you luck but if there is any way you can go with a desktop computer your going to get alot more options and alot more bang for your buck with a self build desktop from buying your own components for $600 bucks.