first off, move from League of Legends to Dota2 (free on steam) it is a little harder but is much better when you get in to it.
Ill source these parts and prices from novatech, although you might find them slightly cheaper if you look around. (but the customer service from novatech is excellent)
4th gen z87 asus mobo
£105
4th gen I5 4670k
£190
G.Skill ddr3 1600mhz 8gb
£55
seagate 1tb 7200rpm hdd
£50
Samsung dvd
£14
Novatech 600w modular psu
£55
how much is that all together? lol
(edit) £468.92.
so 131.08 left to get a case and a GPU (although the built-in gpu on the I5 will play LoL.)
You would need a mid atx case (£50-£80)
Example
And a radeon 7850 hd
£139