There's a few things going on with your situation here.
1) Casting Mystic Reflection targeting the Phantasmal Image will cause the Image's sacrifice ability to trigger first, and whatever the Image was copying will be dead before the Mystic Reflection resolves, which will then fail because it has no targets. So whatever you are attempting to copy won't work.
2) Backfire is an Enchantment - Aura, which can only be cast at sorcery speed, so doing this in response to the rest of the stack sounds very odd.
3) To your actual question: Omnath, Locus of Rage's death trigger will not trigger Backfire. For a creature to "die" mean to be put into your graveyard from play, so at that point Omnath is already there, and Backfire is placed into its owner's graveyard immediately after as a state based effect. And if you had other elementals on the battlefield when Hellion Eruption resolves, their Omnath triggers would happen at the same time, so none of them would be subject to Backfire either.
The Mystic Reflection bit was a mistake, I guess I didnt realize it had defender too, the joys of spelltable. . . .but it was just to try and make the 15 helions I was about to get 5/5's instead.
I played a Hellion Eruption held priority and cast a Mystic Reflection targeting my opponents Phantasmal Image.
In response they cast Backfire targeting my Omnath, Locus of Rage.
I guess the question is with the interaction between Backfire and Omnath's dies trigger. Will the Backfire trigger? Isn't Omnath in the graveyard?
1) Casting Mystic Reflection targeting the Phantasmal Image will cause the Image's sacrifice ability to trigger first, and whatever the Image was copying will be dead before the Mystic Reflection resolves, which will then fail because it has no targets. So whatever you are attempting to copy won't work.
2) Backfire is an Enchantment - Aura, which can only be cast at sorcery speed, so doing this in response to the rest of the stack sounds very odd.
3) To your actual question: Omnath, Locus of Rage's death trigger will not trigger Backfire. For a creature to "die" mean to be put into your graveyard from play, so at that point Omnath is already there, and Backfire is placed into its owner's graveyard immediately after as a state based effect. And if you had other elementals on the battlefield when Hellion Eruption resolves, their Omnath triggers would happen at the same time, so none of them would be subject to Backfire either.
Also my opponent was playing Minn, Wily Illusionist which let him put Backfire into play instead of casting it when I targeted the Phantasmal Abomination. Not a Phantasmal Image