You really should read CT’s article on becoming a strength coach over in the lair.
I personally think that a sport science degree of some sort, while not essential, is a good base from which to work. It will provide you with a good understanding of anatomy, physiology, biomechanics etc. While they tend to lack in the conditioning methodology department you can make this up on your own.
One of the biggest rasons that I think a degree is a good place to start is that it will give you enough of a knowlege base to think critically about information that you are exposed to. On this site you will find conflicting information from different sources, with the aforementioned background you will be more able to make informed decisions. Keep in mind that conflicting info on this site is fairly rare but out in the real world whoa boy!
I have a sport science degree (BPE) so I might be a bit biased but when people ask for my qualifications they usually stop me after I mention that one. It’s something the general public knows and recognizes, unlike the rest of the alphabet soup that follows my name.
Like I said, I don’t think it’s necessary but I think it will help. Remember though success is in no small part due to who you know not what you know. However, being good at what you do and being successful at what you do are two totally differnt things, keep that in mind.