Wow. The H60, if you are on a budget, it's ok but kind of against the idea of budget lol. On a strict budget, I don't recommend much looking at intel. You can get a Pentium G4650 that's a dual core with hyperthreading. They used to be good value, but they recently jacked the prices to 75 or so it seems from the 55-60 they used to cost. Those perform about the same as the i3, so not really a reason to go upward. As far as upgrades, intel is releasing Coffee Lake next year, and it looks like those will need new motherboards. So honestly I'd hold off on intel.
That said, on a budget, where I would want to be is this.
Ryzen 1200-$110
https://www.newegg.com/Product/Product.aspx?Item=N82E16819113446&nm_mc=AFC-C8Junction&cm_mmc=AFC-C8Junction-PCPartPicker,%20LLC-_-na-_-na-_-na&cm_sp=&AID=10446076&PID=3938566&SID=
ASRock AB350 board(this is the no frills no nonsense version but should allow some overclocking)-$59.99
https://www.newegg.com/Product/Product.aspx?Item=N82E16813157763&nm_mc=AFC-C8Junction&cm_mmc=AFC-C8Junction-PCPartPicker,%20LLC-_-na-_-na-_-na&cm_sp=&AID=10446076&PID=3938566&SID=
8gb ddr4 3000 ram(2x4gb sticks---check the qvl on the board to be sure it's compatible)-$75.99
https://www.newegg.com/Product/Product.aspx?Item=N82E16820313769&nm_mc=AFC-C8Junction&cm_mmc=AFC-C8Junction-PCPartPicker,%20LLC-_-na-_-na-_-na&cm_sp=&AID=10446076&PID=3938566&SID=
Power supply-EVGA 450B(may want to upgrade later, but on a budget, it's hard to argue with)-$34.99
https://www.amazon.com/dp/B01F5LX55K/?tag=pcpapi-20
Case-Rosewill Nautilus-$29.99
https://www.amazon.com/dp/B01IT0TDY6/?tag=pcpapi-20
That case actually looks nice for 30 bucks.
If my math is right, about 310.96. If you want an optical drive, count on about 15 to 20 dollars tops. Wouldn't normally recommend an H60 in a budget build, but you know what, run it and overclock the 1200. Maybe opt for a board with more features, but gives an idea what you can do. Don't forget a copy of Windows 10, add another 100 or so for that.
But if you got that 1200, and could put it to 3.7 or 3.8ghz, I'm guessing with a 1050 ti, on a budget, you'd have a nasty little machine that you could later drop a 6 or 8 core into, maybe later upgrade power supply and put a better card in and roll. But for 1080p, that card you've got is fine.