Could it be,
Robert Downey Jr.
Little known I was before,
Now so many do me adore.
Robert Downey was little known before his role as Iron Man
I changed my life around you see,
Now puzzle out who I may be.
He had a lot of problems before his role as Iron Man which I think he said one time saved him. I don't have a link for that just going on memory
I shot that kid, the cover up.
US Marshals, it's the only one that I can see a possible cover up in but I don't get the kid part
Not quite rain, can't fill your cup.
I puzzle it out with my mind,
That explosion could have made me blind.
The nuclear explosion from the avengers is bright enough to blind someone
That man tried to start a war.
Electricity can make me sore.
One of Tony Stark's enemies in Iron Man 2 used electrical whips. I feel like that would make someone sore
Then my friends finally arrived.
Less Than Zero
I don't like my dad, you soon derived.
I don't think he likes his father in the movie The Judge
Plagued by dreams, house blown apart,
Maybe that plan wasn't so smart.
His house was blown up in the third Iron Man
I'll be back soon, don't you pout,
Raise the cry, give a shout!
Captain America: Civil War will have Robert Downey Jr. in it in 2016