1080p 60fps+ PC build under $610

This might be a little tricky to pull off. You'll likely be spending ~$100 USD on the monitor alone, making your effective budget $510. If you go really cheap, you can get hold of a cheap keyboard for ~$10-15.

The Windows 10 license would be a big budget hit. We can get around that by having you join the Windows Insider program and downloading their free alpha/beta build of Windows 10 to use instead. You'll want to move yourself to their slow update ring though, so your experience isn't too buggy.

As far as the hardware itself for the PC, we should be able to work out something with a $500 budget. We're going to need the following pieces of hardware:

1: Case

2: PSU

3: Motherboard

4: CPU

5: GPU

6: RAM

7: HDD/SSD

To begin with, we need to pick out a PSU that's very cheap, but won't spontaneously explode later down the line. I think for that purpose something like this https://www.newegg.com/Product/Product.aspx?Item=N82E16817139027&cm_re=psu-_-17-139-027-_-Product would do well. This bring our total price so far for the PC to $55.00.

Next, we should pick out a motherboard. https://www.newegg.com/Product/Product.aspx?Item=N82E16813132871 This one is on sale for $45. This puts us at $100 total for the build.

The CPU is going to be a bigger hit to our budget. Since the objective is gaming, an i5 would be ideal, but we'll have to settle for an i3 here. https://www.newegg.com/Product/Product.aspx?Item=2MN-0004-00002&cm_re=Core_i3-_-2MN-0004-00002-_-Product We're now at $220.00.

https://www.newegg.com/Product/Product.aspx?Item=N82E16820148858 Which brings our budget to $275.00

Next is probably going to be the biggest budget hit so far. We need to select a GPU capable of driving 1080p at 60 FPS. We're going to have to take the 3GB edition fo the 1060 to pull this off. https://www.newegg.com/Product/Product.aspx?item=N82E16814487263 This brings our total expenditure to $464.00. This is pushing us extremely close to our limit. We have $36.00 left to somehow get a case and an HDD/SSD.

https://www.newegg.com/Product/Product.aspx?Item=9SIA9TS3N29982 this unfortunately pushes us $1 over our budget and is relatively low in capacity, but it will have to do.

This means we have $0 left for a case, so we're going to have get crafty or forego a case entirely. There are tutorials online for building a PC case out of cardboard or other materials, but I recommend wood if you have any carpentry experience. Otherwise, a case would be about $35.00 https://www.newegg.com/Product/Product.aspx?Item=N82E16811353061

I strongly recommend increasing your budget by about $200 for a budget gaming build, but if your budget is absolute unchangable, I would suggest looking around for parts on websites such as eBay or other comparable places for a good deal. A lot of times, people sell "broken" 1080p monitors for next to nothing on these sites, and all they need is a quick capacitor change, if you feel comfortable soldering.

Finally, if you want to get all the bells and whistles without any compromise, holidays will offer discounts on a lot of these parts. Just look around and see what you can find!

Good luck!

Edit: Forgot to mention that cardboard has a flash point of 450F (232.2 Celcius), so it shouldn't burn from your PC components, which shouldn't even go above 80C, let alone 232C. I would keep the PSU outside of the cardboard though, as I have known them to fail in violent and charring ways. I strongly recommend just finding another $35 for a case.