Well an HD 5770 is a bit old, anyways what resolution will you be playing at? If at high resolution then the 5770 would be weak. I myself have the HD 7770 (basically a newer HD 5770 that is more efficient) and I can max out skyrim with ultra settings at 1600x900.
HD 5770 will do fine at 1280x780 (or was it 1280x1024?), however if you can try and ask your friend to sell the computer without the HD 5770, just go buy a HD 7770 and it will max out all games at the resolution.
For 200 pounds your "friend" is robbing you...the 5770 used maximum of 30-40 pounds, to give you a idea, the 7770 which is 2 generations ahead and more powerful is at 100 pounds, with still less than 200 pounds you can get a 7870 2gb ghz http://www.amazon.co.uk/Sapphire-11199-00-20G-HD7870-GD...
Every DirectX11 compatable graphics cards can run every game at maximum settings.The thing is that not all of them can run them at max with high FPS.
And while i do agree that the HD 7770 will run games very well at that resolution, there are games that it can't even play on high @ that resolution.We're not talking ultra here.
And there is also one game that i assume it cant play with more than 40 FPS avarage even on its absolute minimum settings(Crysis 3 Alpha MP).
And now to answer the quistion.The HD5770 is a decent card but don't expect wonders.It will play most games on their high or medium settings with very satisfiyng frame rates but dont expect wonders.
I've got a HD5770 too and i used to play Skyrim @ 1080P on high settings, highly modded with constant 60 FPS.It looked better than ULTRA for the most part but it required a lot of tweaking and optimisation to get it to work this well.
Basicly you'll be able to play Skyrim on High or Ultra settings and get decent frame rates.And yes you'll be able to play League of Legends on its highest settings with a minimum of 60FPS.
(The HD 7770 is around 30% faster than a HD 5770)