I played as a dc and my ds is beginning to get into it, although he's older (7yo).
I've done tournaments (me not ds) and met every type of person from the child who came out of school so they could play/be coached 8 hours a day (and no they didn't make it) and the person who thought they were brilliant because "you know I only decided to start playing last week, but I'm really good" it showed, and no they weren't, but they did have nice tennis raquet earings
It really is as much as you want to put in. I knew people that were happy to have an hour's coaching and an aranged game every week, and people who played 52/7. The latter were on average better, but I'm not sure they necessarily enjoyed the game as much. 
Tennis mums can be lovely, but there are some awful ones out there too. I knew ones who were banned from the grounds when tournaments were on. Then there was the kind dad who refused to give his ds aged 16yo a lift home after he'd lost one final. They lived 40 miles away. 
I wouldn't ramp up the lessons at present, and probably not for at least 3 years, and then only by a small amount if the coach feels it is a good idea. You'll end up with it being a chore and a bore. If you want to help in other ways then swimming will help general fitness and dancing is good for footwork. A child can be overcoached.
Keep the emphasis on being fun. If he wants to mess around with a raquet and ball at home, then that will help his ball skills as much as putting extra time on the court at this stage. If you have a flat wall he can hit against, but if you don't, he can always bounce the ball on his raquet round the garden or park.
In all honesty if you have a good coach, if he's showing aptitude then they will approach you rather than you them. A good coach will pick up on the good ones because they want to have them representing the club/the coach which advertises them (and is very satisfying personally too).
I'd say you want to look for a club that does junior events. Not just has a competitive squad (as a rule about 90% of parents think their dc will sail into that, even when it's about 25% actually get in) and coaching. But social days for juniors (although that's probably older) or social evenings (often called club nights) or fun tournaments.
And be wary of the school one. I've not yet seen coaching (even run by acredited company) going on at a school grounds where I haven't ached to wander through the children correcting grips, and giving very little pointers that should have been given much sooner and would make huge differences to their play. It often seems to be a bit of a "well we wave our rackets and if we hit the ball it's a bonus" free for all.