Yes, you can mine with a normal PC 
But no use mining Bitcoin directly as others mentioned, you can still change altcoins to Bitcoin if you want.
I started with Litecoin mining on gpu, then got a hint about just released Protoshares from a irc-friend (inside nerd channel but I was the only one who started) and started mining it with cpu at same time. When it seemed that Dogecoin is here to stay, even some months, I changed to it from Litecoin with gpu, and when Metiscoin was published I mined it with cpu until difficulty was too high. Now I'm mining Protoshares with cpu and Dogecoin with gpu again, but when something interesting appears I will take the easy coins without hesitation. That's the interesting part on this hobby, always searching for something new that could be the next big thing
Oh yes, I have i5 laptop too which have been mining protoshares nearly from start. No enough gpu power for mining, it has the slowest possible Nvidia chip, and cpu mining speed is a bit less than half of my desktop computer's cpu power (Ivy Bridge i5 4,2GHz).
But no use mining Bitcoin directly as others mentioned, you can still change altcoins to Bitcoin if you want.I started with Litecoin mining on gpu, then got a hint about just released Protoshares from a irc-friend (inside nerd channel but I was the only one who started) and started mining it with cpu at same time. When it seemed that Dogecoin is here to stay, even some months, I changed to it from Litecoin with gpu, and when Metiscoin was published I mined it with cpu until difficulty was too high. Now I'm mining Protoshares with cpu and Dogecoin with gpu again, but when something interesting appears I will take the easy coins without hesitation. That's the interesting part on this hobby, always searching for something new that could be the next big thing
Oh yes, I have i5 laptop too which have been mining protoshares nearly from start. No enough gpu power for mining, it has the slowest possible Nvidia chip, and cpu mining speed is a bit less than half of my desktop computer's cpu power (Ivy Bridge i5 4,2GHz).