You can combo pretty much anything out of archfire, but there are a few I use more often then others. I will list them here. Let me just list all the ones I can name off the top of my head because I get easily carried away.
-Archfire into Up Smash: This is my bread and butter. Archfire carries the opponent upward, and Up Smash comes out REALLY fast and kills at very respectable percents. It also covers a wide range if they are trying to mash out of it, and you would be surprised the range you can do this combo out despite your crap ground speed.
-Archfire into Back Air: This is your second best option and is almost just as good as upsmash. Use this when you want to kill your opponent off the side, or you want to knock them off stage. I most often use this when my opponent is at the ledge and there simply isn't any room to do up smash. It kills ridiculously early if used at the edge of the stage, I'm talking like 80-90% here.
The only downside to this option is it takes a bit of practice to pull off consistently. Go into training mode and practice this one every now and then, I myself have gotten quite rusty at it after my break!
-Archfire into Archfire into Up Air or Up Smash (If the stage has platforms): This is your best option at low percents as it is when the combo is the most consistent. Go for it whenever you land a direct hit with arch fire as this will stick your opponent into the center of the hitbox where they cannot mash out. Once again, this one takes practice. Though it is absurdly easy to pull off against heavies, so always go for it against them.
Just remembered, sometimes heavies are so heavy that you can simply throw another Archfire at them without even jumping. This makes up smash easier, or you could try going for triple Archfire.
-Archfire into Thoron: This is your best option if you cannot reach your opponent for some reason, or if your archfire catches the opponent hanging off the ledge. Kills really early because well.... it's Thoron.
-Archfire into Down Smash: Used in the exact same situation as Thoron, serving as a substitute when Thoron is not fully charged. It is overall less reliable, but it still works.
-Archfire into Nosferatu: This one is very situational, as the only time this combos reliably is usually after a direct hit, when another archfire is often better. Use when there is a major percent difference between you and your opponent, or if your opponent is too high up to land the second Archfire
Archfire into Up air or Forward Air: Lastly, this is probably one of your less optimal options, as any time you can land this you can probably land a back air instead, which is almost always more optimal except for at 130 + percent where it doesn't really matter what aerial you hit them with.
This is basically what you go for when you were not expecting your opponent to get caught in your archfire and you need a quick reaction, or if your opponent is too high up and you don't need the boost from Nosferatu. It's still a good option, and deals good damage, it's just not optimal.
And this fellow tacticians, is one reason I love this character XD. Between Thoron and Archfire you have so many improvised combos and traps! The last thing I will say is that Archfire makes for excellent shield pressure, so if your opponent blocks it, feel free to throw out another one! This scenario is completely safe from my experience, especially if they are at ledge!