I know Minato (the fourth Hokage) is of the Namikaze clan, but I have never heard of anyone other than him belonging to the same clan. What is the Namikaze clan's history? Is it part of the Senju clan?
Namikaze isn't a descendant of Senju. I think it would have been mentioned in the past if it was, just like how it was stated Uzumaki is related to Senju. The past references has been brought up so many times with Kushina having Senju DNA, and therefore Naruto has it too.
As for all the members, we only know of Minato. No other Namikaze member was revealed in the manga. I haven't watched all the fillers, but there might have been another Namikaze person in there.
The Namikaze clan is well know for their speed, as seen by the most well known Namikaze's Minato. Despite his extensive use of the Flying Thunder God Technique, his speed came natural. Members of the Namikaze Clan are gifted with high level of intelligence. Due to their speed and ability to master any or almost all ninjustu, they became a nearly invincible clan. They were even capable of standing up against the Uchiha and Senju to a certain extent. Prior to Minato becoming the Hokage however, this clan gained no recognition.
There are some other members mentioned in this page.