i5-4250U is a laptop processor.
If you buy the desktop version of the i5, you'll have to buy a motherboard and ram too, you can't just buy a cpu and put it in your old board.
Yes i5 is quad core.
I think your very confused, you should post asking for suggestions by listing your full computer specs including the make and model number of your motherboard and the type of ram you have, and asking how you can upgrade it.