For Bloodcrazed Goblin to be able to attack, an opponent must have been dealt damage that turn before the declare attackers step begins. The damage must have been noncombat damage of some sort, or it was combat damage and an additional combat phase has been created.

Bloodcrazed Goblin’s ability checks only whether damage was dealt to an opponent. It doesn’t care whether that opponent also gained life. For example, if an opponent was dealt 4 damage and gained 6 life during the turn, that player will have a higher life total than they started the turn with — but Goblin Berserker will still be able to attack.

If damage would have been dealt to an opponent but it was all prevented, redirected, or otherwise replaced, Bloodcrazed Goblin won’t be able to attack.

It doesn’t matter how an opponent was dealt damage, or even who controlled the source of the damage.

If Bloodcrazed Goblin attacks, it doesn’t have to attack the opponent who was dealt damage. It can attack a planeswalker. In a multiplayer game, it can attack a different opponent.