Since your budget is 300$ I would strongly advise going with second-hand parts.
Depending on where you live, a decent cpu+gpu can be well over 300$.
If you have already have an old PC you can salvage your current case to save ~20$ bucks.
Also @dimooz
[quote=dimooz]Amd ones are cheaper but need better coolers than Intels. I’ve got a Phenom II 4 x 3,2GB, oc to 3,6GB, that works really fine.[/quote]
You mean GHz instead of GB?