If I Have Bizzarro as my supervillian, when my opponent or another outside source causes me to destroy one of my cards do I still gain a weakness?
It depends on what the card says.
Broadsword, for instance, says "Destroy a card in another player's discard pile.." In that instance, even though it's your card getting destroyed, you're not the one doing the destroying, so you don't get a weakness. Similarly, if you're the one using Broadsword on an opponent's card, you did the destroying so you get the Weakness. This also applies to Superwoman, Ultraman, Power Ring, etc destroying cards in the line up.
Most of the Super Heroes' first attacks say "each player destroys..." which means you're the one doing the destroying, even if it's not a source you control telling you to destroy it.
That clear it up?