To get this, you'll need to have completed all 17 missions on hard or nightmare difficulty. Fortunately, you do not need to do this in one playthrough from start to finish; you can go back and redo levels on harder difficulties. In the chapter select, you'll see by the level name the hardest difficulty you've completed the stage on.
As for accomplishing the task itself, the easiest solution is to have more skylanders, as the more you have, the more meat you can throw in the grinder. But, for some, the concept of buying more figures disgusts them, leading to the question as to why they got the game in the first place, but anyways:
Pretty much the main difference between difficulties is how much health the enemies have and how much damage enemies and hazards do to you. On hard, most hits will do around 150 damage, meaning that even a maxed out skylander can't take more than a half-dozen hits or so before dying.
If you want to survive, you need to keep moving. Stand your ground as little as possible and focus on hit and run tactics to whittle the enemy down. There's no time limit, so patience is key for survival.
If you are just using the starter figures, you'll want to level them along certain branches:
Stealth Elf should be your main melee fighter, taking the blade upgrade path. When fully upgraded, she has insane DPS, and can dispatch golems with relative ease. She is a bit fragile though, so make use of her acrobatics and her stealth to escape enemy attacks. When facing a large group, make use of the decoy to draw some of the enemies away, then fight the half that went after you (Divide and conquer and all that).
Wash Buckler will be your main ranged unit, so go for the upgrades to the bubble attacks rather than the cutlass. Wash will be your best bet for larger groups as when upgraded, his bubble gun will trap enemies, and a trapped enemy is one that's not trying to kill you, so keeping everybody trapped while the piranhas devour them is the way to go. He can't trap golems though, so watch out. As for the bottom half, neither is really better than the other as the upgrades are for attacks which require you to stand in place, which is not a good thing in the harder difficulties.
Blast Zone will be for more chaotic fights. Upgrade your bombs and chuck them everywhere, letting them ricochet into enemies. You'll also play keep away by dashing away, then setting the ground on fire, letting the enemies run into it and get trapped by the flames, whereupon you'll shower them with bombs. For his bottom half, neither is particularly more useful then the other, so it comes down to personal preference.
1) Prioritize your enemies. Take down those which will give you the most trouble first. Generally Spell Punks would be top priority (especially life ones) followed then by ranged units. After that, it'd be whoever tends to be hardest to avoid.
2) Swap out your skylander when it's health is low. A defeated skylander can't be used for the rest of the stage, so it's best to pull them out until you can find health.
3) Use Elemental gates as rest stops. If you open an elemental gate, all skylanders of that type will slowly heal while inside. So it's beneficial to just sit in there and soak your wounds for a bit.
4) Make use of Charms and Legendary Treasures. Basically, each charm and legendary treasure you place on a pedestal will power up your skylanders a bit. It's not much, but any help is better than no help. For the treasures, try to use ones that give max health or crit chance, as armor won't mean much and elemental power only takes effect in areas your skylander is marked strong in.
5) Don't fight everything. Most fights are skippable if you just simply run past the enemies and onto the next level. Otherwise, if you drag yourself into each fight, that just means more opportunities to take damage or worse, lose a skylander
6) Use a weaker Skylander for rail segments. Basically, any time you take a turret or do some rail grinding, use one of your lesser used skylanders. This is because what skylander your using has no bearing on that segment and also if you mess up, it's not your star fighter that takes the hit.