How much damage do I deal? (How is damage dealt if an attacker gains power as combat damage is being assigned?)
You will only deal 6 damage to your opponent.
This is because when determining how combat damage gets dealt the fist thing you do is assign damage based on the power of the creature (in this case 1 gets assigned to the blocking creature and 6 to your opponent), then after you decide how it is assigned it actually gets dealt. The fact that the creature gets bigger at the same time that it is dealing damage doesn't matter since it can only deal the damage it has assigned.

510.1. First, the active player announces how each attacking creature assigns its combat damage, then the defending player announces how each blocking creature assigns its combat damage. This turn-based action doesn’t use the stack. A player assigns a creature’s combat damage according to the following rules:
510.2. Second, all combat damage that’s been assigned is dealt simultaneously. This turn-based action doesn’t use the stack. No player has the chance to cast spells or activate abilities between the time combat damage is assigned and the time it’s dealt.