Can I deal damage to the opponent’s health after I kill their creature trying to block my attack?
No, if an attacking creature is blocked, then it can only deal damage to the creature(s) that are blocking it. (Unless the attacking creature has trample.)

After attackers and blockers are declared, all attacking and creatures deal damage. Your 5/5 will deal 5 damage, and if it being blocked by a creature, then it must deal all 5 damage or that creature. So you will deal 5 damage to your opponent's 3/3, and at the same time your 5/5 will take 3 damage from the blocker. Then the 3/3 will die because it has more damage than toughness.

If your 5/5 had trample, then you could have assigned 3 damage to the blocker and the other 2 to your opponent; killing the blocker and dealing 2 damage to your opponent.

Also, you say “I kill the 3/3 creature because I have more toughness” but this is incorrect. You kill the 3/3 because your attacker has 5 power, which is more than the 3 toughness that the blocker has. 

From the basic rules:


  Once all blockers have been chosen, each creature—both attackers and blockers—simultaneously deals damage equal to its power (the number on the left side of the slash in the lower right corner of the card).
  
  
    An attacking creature that isn’t blocked deals damage to the player it’s attacking.
    
    An attacking creature that is blocked deals damage to the creature or creatures that are blocking it, and vice versa.