I have almost the exact same setup. i7 920, HD 4870 1GB GDDR5, and 12GB of DDR3. I say with a fair amount of certainty that it's not possible to render off of this GPU. According to Blenders documentation GPU computing for AMD/ATI is only supported on the GCN architecture. In the video when he enables GPU compute, his GPU is listed as Pitcairn.
Pitcairn is a 28nm GCN1 chip from 2012, where as the HD 4870 is a 55nm chip from 2008. Not that fab size matters, but it just goes to show how old each series is. In case you were wondering, the HD 4870 is on the TeraScale1 microarchitecture. GCN only started with the 7000 series. Essentially all discrete desktop AMD/ATI GPU's 77xx and up will work. Anything lower, like the 4870 just won't. Don't believe me? Try searching for results on Blenchmark. For those wondering about desktop Nvidia cards it's a bit more convoluted, but here's a list.
NVIDIA TITAN X, GTX 1080, GTX 1070, GTX 1060, GTX 1050, GTX TITAN X, GTX TITAN Z, GTX TITAN Black, GTX TITAN, GTX 980 Ti, GTX 980, GTX 970, GTX 960, GTX 950, GTX 780 Ti, GTX 780, GTX 770, GTX 760, GTX 750 Ti, GTX 750, GTX 690, GTX 680, GTX 670, GTX 660 Ti, GTX 660, GTX 650 Ti BOOST, GTX 650 Ti, GTX 650, GTX 560 Ti, GTX 550 Ti, GTX 460, GTS 450, GTX 590, GTX 580, GTX 570, GTX 480, GTX 470, GTX 465, GT 740, GT 730, GT 730 DDR3,128bit, GT 720, GT 705, GT 640 (GDDR5), GT 640 (GDDR3), GT 630, GT 620, GT 610, GT 520, GT 440, GT 430.
*The GT 545, GTX 645, GTX 745, and GT 710 all appear to work despite not being listed on Nvidia's site. You can also find my source plus supported mobile GPU's in the link below.
Note: Going back to the original question, just for some context, the Radeon HD 4000 series was competing against the GeForce 200 series which also doesn't support GPU rendering. It was two years before the release of the GeForce 400 series, the first to support GPU rendering. The card is just to old.