If you get hit, you can exit to the lobby and then continue the same run from its last save point. It's is still a difficult achievement, but not losing all of your progress makes it much more doable. If you aren't sure whether you've been hit (if you have some healing ability), just watch your combo counter.
I got it after using laser cannons (the dual laser pistols). In many levels, their infinite range allows you to kill enemies while they can't even reach you. Melee doesn't matter too much, but I like Claw. Knockback on nearby enemies helps keep avoid getting hit by a random crab. I brought the airstrike ability which helped clear out masses of enemies and early bosses, but I focused on upgrading my main weapon.
Also note that despite the achievement being for not "taking any damage" it's actually more strict. Getting hit while you have armor will prevent you from getting the achievements. Some reddit comments suggested that losing max health (like from a greed perk) also prevented it.