Unfortunately you need a new CPU, motherboard, and memory.
£300 for the chip (CPU) won't even get a Skylake i7 (the i7-6700K is £320), as prices have gone mental. The i5-6600K is much cheaper at £195.
This is why people are saying if you're going for an i7 you might as well get a Haswell-E i7-5820K for £300 which has six cores.
On the other hand though, the cheapest X99 motherboards are a lot more than Skylake - £170 for an X99 motherboard - compared to as low as £90.
But if £300 includes CPU, motherboard, and memory I'm afraid the current gen is out of reach, as an i5 bundle would start at about £330.
Edit: alternatively you could wait for the lower range Skylake chips to come out, or look at a previous gen system.