How did Itzhak Perlman become so good with the violin? Was it his
parents, etc. who were musicians? Were they? I can't find this
information anywhere; I can only find information AFTER he went to
America, then it starts there (I also just know he was from Tel Aviv,
and the music school that he went to.)

Please tell me more about his childhood, though, thanks!

Like many musical geniuses, Perlman showed an interest in music at an early age:

"He began his career at the age of 3, when he asked his parents for a
violin. The next year he came down with polio. Ever since he has had
to wear braces on his legs and walk with crutches.

In 1958, when he was 13, he performed on The Ed Sullivan Show."

http://www.cbsnews.com/stories/2000/01/17/60II/main150331.shtml

Here's an interesting interview with Perlman in which he talks about
child prodigies:

http://www.finelinefeatures.com/shine/news/nytimes.htm