Your system is about the same generation as mine, second generation Core iX series with an LGA-1155 socket, that H61 board is a Sandy Bridge board. Which means you may run into BIOS compatibility issues. Last BIOS update was in 2012 -
http://www.gigabyte.com/products/product-page.aspx?pid=3773#bios
You could probably (and that's the key word, probably) run something like a GTX 900 series or an R9 series card, you could almost certainly get a GTX 700 or 600 series or an R7/HD 7000 series GPU working.
You may, or may not get an RX 400 series or GTX 1000 series card to work on that board.
If you can't afford a new PC with a Sandylake or Kaby Lake processor (6th or 7th Gen Core iX CPU), minimum upgrade would be to an Ivy Bridge motherboard with a z77 or maybe q77 chipset, and a BIOS update that is as new as possible. Asus, for example, was kicking out BIOS updates for some of their z77 boards into 2014.
The newer you can afford to get, the better. And considering some of the prices I'm seeing, a new i7-6700 or i7-7600 chip is CHEAPER than my dinosaur of i7-2600. Go figure.