"I have known him since his childhood" is correct. With the present perfect tense, "since" indicates the beginning of the period of time that the speaker is referring to, here, "his childhood."
There is a way to use "from" with "know." Here's an example of a group conversation:
A: How do you know Max?
B: Everybody knows Max! I know him from my history class.
C: I know him from my dorm!
D: I know him from Englewood, New Jersey. We went to high school together there.
E: I know him from the homeless shelter. We both volunteer there.
F: Oh, Max? Is he here at this university? I knew him from our high school basketball team, but I've lost track of him.