He has lots of options, the tricky part is knowing reading your opponent. Some people will adapt to things, others won't, so you have to feel it out on the fly.
These are the moves I use to kill:
Down Smash - This is probably my preferred kill move, but the only situation I use it in is edge guarding. Stand on the ledge after you've knocked someone off and unleash this when they try to grab the ledge. It reaches out far enough and down low enough that it can usually guard the ledge pretty well. You may need to hop out and hit them with an aerial first though, just to waste their second jump, then you can use this move to counter their Up Special.
Up Smash - A little easier to catch someone with on stage than the Down Smash. Still, I basically only use this when someone is in the air above me, not so much for when we're both on the ground.
Side Special - Contrary to what some other people have said, I think the spinout is great as long as you time it right. I'm rarely punished for it. But then again I use this move sparingly enough that people rarely see it coming. It moves so fast that if they do react it's usually just to block and by the time they think of counterattacking I'm already ready again.
Side Tilt - This move has some decent range and is fairly safe compared to a lot of Bj's options. It can kill at higher percents, but I mostly use it for safely poking and trying to push people off the stage.
Forward and Back Air - Both are pretty solid if you get the timing and spacing right. The opponent has to be a little ways away from you. They can be useful for killing at higher percents (lower than the side tilt though) but as stated above, I mostly use them for keeping an opponent away from the ledge or racking up damage.
Neutral Special - Not going to kill at low percents, and by the time it will kill I feel you have better options, but if you want to go for style points you can launch one of these at a foe who's off the screen for a cheap kill.
Up Special - The exploding cart can kill, but the timing is tricky. Once you try it a few times you can decide if it's a move you want in your repertoire or not. Similarly, Bj's hammer while he's using this move is actually pretty good if you can get the timing right.
For edge guarding (which, like I said, usually just ends in a Down Smash) I'll use these:
Down Air - This has, on some occasions, just the slightest spike to it. Not enough that I would attempt it in a tournament or anything, but just for a CPU battle or something it's fun to mess with. It's spiked (and I use the term loosely, but it did double their fall speed basically) for me a few times when I've just barely tapped opponents with the tip and only with one hit. If you graze someone with the side and do some but not all of the hits then it can be used to gimp their jump/upward momentum or something and just leave them falling. If you hit with the last part of it though then it's going to knock them away in kind of a diagonal upward path. It's a good move for racking up damage and even a bit of keeping them away from the edge and stuff, but I wouldn't consider it a kill move.
Neutral Air - Good horizontal hitbox and it comes out quick so you can use it pretty fast on the ground with short hops and such too. Another one that I don't use for direct kills but I use a lot for spacing and such.
Down Tilt - Like the Down Smash I use this a lot while teetering on the edge to intercept people who are about to grab it. I use it when I know there's no chance of a Down Smash getting a kill (like if they're at 30% or something) but I want to get some damage while keeping my Down Smash fresh. This also comes out pretty quick so on the ground I'll use it as a punish, similar to the jab combo.
And these moves I feel you should be careful with:
Side Smash - This is a strong move and if you're in a situation where you're 90% sure you can catch a falling opponent with it then go for it, but I wouldn't advise throwing this out randomly. It's rather slow and if you miss you're wiiiiiiiide open.
Up Tilt - This is actually a pretty good move for when someone is above you. It's quick and can combo a bit early on. But the hit box is kind of deceptive. I've had people standing directly in front of me and watched this move whiff over their head. It seems very situational to me.
Dash Attack - This is a decent move for wracking up damage and catching people as they're about to land. If you can catch someone off guard with it it works nicely, but use it sparingly cause it can very very easily be blocked and punished.
Bowser Jr is currently my favorite character in the game and I actually haven't lost with him yet in For Glory 1v1s. You just have to adapt to your opponent. For instance my Down Smash on the ledge kill probably won't work against someone like ROB who will just recover a mile above your head, but repeatedly jumping off the stage and knocking him away from it is a breeze. I hope this helps you at least a little though!
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