I've been messing around with Bowser Jr. a lot. I use the Roy skin, since Roy is my secondary character in P:M. :^)
He's a very satisfying character to use! All of his moves feel strong and have nice hitboxes -- there isn't a single normal or special move that I would call useless. His throws are a little lackluster though, since they don't really get early KOs, they don't set combos up, and they don't do a lot of damage. Up Throw can be used to start a juggle, but it still isn't much, really.
Like Katakiri said above, the extra jump out of Side B is so useful! I've had lots of success by footstooling my opponents after a successful hit, then chaining a DAir. It's unexpected and racks up a lot of damage. The extra jump on recovery is also something not many people seem to be aware of yet, making Jr. really hard to KO.
On the topic of KOs, this is where I'm having kind of a hard time. Jr.'s most powerful KO move seems to actually be the exploding clown copter from his Up B; I've gotten >90% kills on some characters. However, it's pretty unreliable, and leaves you without your main vertical recovery option. However... the hammer move you get from being out of the copter is also really strong. Sometimes I use Up B just so I can get the hammer attack. Up Smash is also a strong KO move, but to be honest I'm not all that used to using smash attacks on the 3DS analog thingy.
All in all, I really like Bowser Jr. If anyone has any questions and/or requests about him, I'll be glad to answer.