NOTE: If the phone is new, recently purchased, I would take it back and get them to exchange it.
It could be a few things. The cord could be damaged. The port where the cord connects could be also. Then the battery could be dead/no good.
Did you try leaving it charging for a few hours, rather than just a few minutes?
Also, you might try holding the power button for around 10-15 seconds, then trying to charge it.
Another option, something is stuck in the power button, lint, dust, food, etc., making it stick and now allowing it to turn on (this can happen with any of the buttons).
If none of the above work, I would really suggest a new cord and/or a new battery. Should those not do the trick, then you may need to either have it repaired, or get a new phone.