If I purchase bitcoins via p2p, can I get some problems if the bitcoins I get were involved in some criminal activity ? If yes then how do I avoid such situation ?
I don't think this will get you into problems, if you deposit those bitcoins to any exchange they will not block your depo. And if you are still paranoid then you can use a Mixing service to Wash your coins.
Another fact to consider is the amount, if you do this with a small amount then there will be no problems, but if you do it with a big amount then the alarms will sound.