It's not really true any more that games only use one or two threads - modern games frequently use 4 or more threads effectively, and in fact some games struggle if you only give them two cores to run on, which is why something like a Pentium or i3 is a poor choice for a gaming system.
With the 6300, you get 6 cores and excellent muti-threaded game performance for £100, and it's probably what I'd recommend given a £300 CPU+GPU budget gaming system. If you're willing to spend 50% more, an 8350 should outperform it in some games, and give you more potential future-proofing as games continue to use more threads. But I'm not sure it's worth eating into your GPU budget for, unless you're OK with upping the total budget.
That 7870 is a Tahiti LE model so more like a cut-down 7950 - it's generally a fair bit faster than a standard 7870 and even comes close to the 7950 in some games, so is great value for money. The downsides are that it's less power efficient, and they are only made by a few brands.