Well you have 2 options. 1 is a new video card, and #2 is a system upgrade.
For a video card since you are only running 1080p I'd get a Radeon RX480 8GB or a GTX 1060 6GB. Both are in the same ballpark price wise. If your monitor was a freesync the RX480 would be the easy choice, but it's just 144Hz so either video card works. Right now you have a Radeon 7700 series so even the RX480 would be a major upgrade. The 1060 is faster in DX11 games, the RX480 in DX12 games.
The next option would be a system upgrade. Hellfire13 posted a half decent upgrade.
Personally I'd think the I5 3330 still works. I'd maybe try to get another 4GB of RAM to get by for now and get the video card to start. An RX480 is around $240 for the 8GB version. The $250 saved can be put aside until you save enough for a full upgrade.
3rd option is crazy but trying to do both. Lets see if it's possible.
PCPartPicker part list /
Price breakdown by merchant
CPU: Intel Core i5-6400 2.7GHz Quad-Core Processor ($174.89 @ OutletPC)
Motherboard: Gigabyte GA-H110M-S2H GSM Micro ATX LGA1151 Motherboard ($60.88 @ OutletPC)
Memory: Kingston HyperX Fury Black 8GB (2 x 4GB) DDR4-2133 Memory ($48.99 @ Newegg)
Video Card: Sapphire Radeon RX 480 4GB NITRO+ 4G Video Card ($224.99 @ NCIX US)
Total: $509.75
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2016-11-12 13:59 EST-0500
So for $500 you can get a full system plus video card. The 4GB RX480 is fine for 1080p. Just some information, the k series CPU hellfire speced is nice but doesn't include a heatsink. The I5 I choose comes with one so it keeps the price down. Intel doesn't need much cooling anyways so the Intel heatsink is plenty. And yes with these new CPUs you need a new motherboard to support the new socket, and also new DDR4 RAM.
There was an RX480 for $20 less but I choose the Sapphire as they are regarded as the best Radeon cards.