Did or do?

Hello

If you wish to ask about something that happened in the past ("got to know"), the right choice is #1.

You could use #2 if, for example, you were asking for advice; if you wanted to know what you'd have to do in order to get to know that person yourself:
"He seems to be a very private man. How do you get to know him?"

There may be other possibilities, but I'm soooooooo late for work now!

Miriam