A creature with double strike will deal combat damage in BOTH the first strike and regular combat damage steps. This applies for whenever the creature is dealing combat damage to a creature, player, or planeswalker.
So if it's unblocked it will deal damage equal to double its power to the defending player or planeswalker. If blocked it will deal damage to the blocking creature twice, unless that creature dies after first strike damage.
Note that if the blocking creature dies after first strike damage, the creature with double strike will not deal damage to anything in the normal combat damage step.
As a reminder for the last thing Arachnarchist said, a creature with both Double Strike and Trample, should it kill a blocker after the "First Strike" portion of the damage, will also hit the opponent in the normal damage step.
For example: If a 3/3 Double Strike/Trample is blocked by a 2/2, 1 damage rolls over during First Strike damage and another 3 will hit the opponent during normal damage.
It's also important to realize that Double Strike doesn't "double" the amount of damage a creature deals. At least not always. If you give a creature like Bloodcrazed Neonate Double Strike (perhaps through Silverblade Paladin) and it hits the opponent twice, 5 damage will be dealt. It hits once, triggers for a counter, gains the counter, then hits again for 1 extra damage than before. Then it would gain a second counter and be a 4/3 after combat, having dealt 5 total to your opponent.
also, any extra "when x deals combat damage" effect will be doubled, so for example if you put Sword of Body and Mind on a double striker and it gets through, you get 2 wolves and they mill 20
So would you gain double the life if you had a double strike life like kill a creature in the first strike combat step?
Unless a creature has both Double Strike AND Trample and it kills a blocker in the "First Strike" damage step, it will not deal damage again in the normal damage step. This is because the blocker is already dead and off the field by the time the second damage comes around, and since your creature was blocked, you cannot deal combat to the defending player at that point.
3/3 Double Strike and Lifelink is blocked by a 2/2. You will gain 3 life.
Quick note: This is all assuming there was only one blocker. A Double Strike creature won't deal damage a second time around if it has destroyed all blockers during the First Strike damage step, and not necessarily just one blocker.
Unless is has Trample, of course.