Dragons make great last ditch or early burst DPS. They've got a crapton of damage. Two melee attacks, a ranged attack at level 3, a short stun at level 4, and a ranged AoE at level 5.
What I like to use them for is their Nom ability. It allows them to outright kill anything they attack under a certain percentage of health, and with the damage they deal, that tends to happen rather quickly.
The trick, though, is that they can't be paired with Ice mages. They cancel out each other's debuffs. So wherever you put dragons, put the Ice mages at the other end. You can still support the dragons with other units, such as the Rangers (which, really, you should be doing anyways), and let them handle either the stragglers or initial wave.
If you're putting them at the front, they have enough health to take several hits while dealing lots and lots of damage. Anything they hit will also continue to burn (and if you put points into Epic Halitosis, they'll take a good chunk of damage from poison, too). This will allow your other forces to mop up whatever makes it through the dragon gauntlet. For efficiency, pair them with a Knight to shatter armor. They'll do more damage, and Knights at the forefront is good planning, regardless
If you're using them as your mop-up force at the back, giving them at least one ranged attack will allow them to do that little bit of extra damage, giving Nom a much greater chance to work. Obviously, the higher the level, the better, but you should upgrade them to at least level 3.
Wherever you put them, though, you want to make sure the Ice mages are WELL away from them. Give the burning time to work, even out of their range. If they're at the end, you don't want the burning cancelling out the slow.
Fun fact for Ice mages: their slows stack, even with themselves. If they slow something, and attack with another ability that slows, it doesn't cancel out the previous one. The time gets added to the existing slow. On the sheep level, you can get the super Sheep to have several minutes of slow on it. If your goal is to maximize damage in this sense, then you're better off not using dragons until the very end. You want that slow to last as long as humanly possible.