Bf4 is a good test
Depending on the card,set the game for high settings & use Fraps for FPS(frames per second).Watch the temps,anything above 65-70C is bad.
Clean the heatsinks inside by either removing the cover (not the heatsink),some have screws,some just snap on & blowing the dust bunnies out or use an old toothbrush.
If you can't remove the cover,use a little flashlight & a piece of wire to fish the dust bunnies out through the fan opening
If you do get artifacts,new heatsink compound & TIM pads may fix it,not always though.I have saved 80% of vid cards this way
I bought a 5870 on ebay about 1 year ago,the fan died 2 months after I bought it so I got a new fan & reapplied the compound & got new TIM pads,still mines scrypt @ 400mh @ 60C with the fan @ 65%,no OC
If you can't save the card,pull the heatsink & fan,ebay them & you can sell the PCB for scrape on ebay,there's gold in those PCB's