There`s this guy that works in the gym who's french as it`s in Paris and I`m quite sure that he fancies me, but sometimes am unsure as when there are lessons, there`s a lot of people there.

I would have to say that French men are very different to English men, they have a different culture over there and that shows with the way they treat women and seduce women.  They tend to be very forward and obvious about their sexuality and their attraction to women, almost like a don juan.  I would have to say that foreign men is one of my weak areas, I have spent a fair bit of time in France and with french males and females but I'm still limited in my knowledge.  So I can't quite provide as much detail in my response or be quite so confident that my reponse is accurate, but I'll try my best.

I get the impression that each time I walk by, he notices changes, and looks at what I`m doing.

If you get that feeling, then I believe you should trust your instincts. 
 

He`s always trying to make everyone laugh and showing off in the lesson.

That's the French way that I always find is common amongst french men, they breed passion, confidence and sense of humour.  In fact as an English bloke, I'm quite jealous of French men, because I'd love to breed all of those elements.

He`s not the one teaching it, he`s just joining in. He never comes up and talks to me and I am very shy, so am too scared to make the first move and wouldn`t know what to say.

Believe me, if he is interested in you and he is the life and soul of the class, it won't be long before he pushes himself towards you.  If I compare french men to a more primitive man, this would be how I explain them.  They eye their prey watching their every move, they impress their prey with any attention seeking method possible, then they move in for the kill, pushing their passion to the maximum.

Since I`ve been there a while he may think that why did I take so long to talk to him.

I doubt it would dent his self-confidence, if he does fancy you, he knows he would capture you eventually.
 

One day he acts like he really fancies me, then the next he acts as though he`s not interested, etc.

That's common amongst people that fancy you, both men and women to display this trait.  I often speculate why men and women do this, one is that they are playing hard to get or don't want to be seen as too keen, because that does tend to put people off.  Another is that you flirt, back off, take in the reaction, flirt again, judge the reaction and you keep doing that until you both meet in the middle.  Thirdly is they have some doubts about whether they are really interested in you.  Fourthly, it's a fail safe mechanism, you seem as though you fancy someone, you are not sure whether the other person is interested, so to save face you pretend your not interested because you don't want to look a fool.  Finally, one of my theories, which is not backed up by research, is the possibility of a change in hormones.  Men's hormones tend to fluctuate once a week, unlike the female's once a month.  As a result his interest in women changes when his hormones are low, he quite frankly can't be bothered making an effort, when they get high again (usually the weekend), he is keen an eager to find a woman, so he can release that build up of hormones. What it is in your circumstance I don't know because you don't give me enough information.

What should I do about this situation as I really like he, and would be happy enough for now just to become friends.

There has to be a time when one of you make a move to make conversation.  All you need to do is be relaxed and signal a friendliness to have a conversation with him.  I must say this guy sounds like he is confident talking to people, as he is confident drawing attention to himself and making jokes to the whole class.  The fact that you have kept quiet, unlike the others who have probably thrown themselves at him, probably means he is more intrigued by you and more intrigued to get to know you.  Nevertheless if you get the opportunity, there's no reason why you shouldn't make some conversation, even just brief stuff like, hello, how are you.
 

I`m sure he fancies me as I always catch him looking my way.

If you get that instinct then I would trust your instincts.