Abilities
P - Phoenix Spirit
Human - Abilities drain 10% of Infurnis's current HP and deal 50% bonus damage
Phoenix - Auto Attacks deal 10% of Infurnis's AP in magic damage and heals him for 50% of the amount of magic damage done.
Is he melee or ranged in these forms? Probably a useful thing to tell us.
Also, I'll remember the 50% bonus damage: that'll be important when number crunching. The healing isn't that important: even with 600 AP you're only dealing 60 bonus magic damage with your auto attacks and healing for 30 per hit. That's not high enough to be concerned about, especially not in the early game.
Q - Flaming Shot/Scald
No Cost / 50/60/75/80/85 Mana(7/6/5/4/4 CD)
Human - Does 100/175/200/225/250 and 70% of AP to target in magic damage and explodes, dealing half the damage to nearby enemies.
Phoenix - Does 100/100/150/200/250/300 magic damage and 50% of AP in physical damage and heals for 20/30/40/50/50% of the damage done.
I assume the mana cost of No Cost / Mana Cost means that the Human form has no cost to the ability, and the Phoenix form pays mana.
So let's look at this. First things first (and I have to tell everyone this): scaling in League is consistent across levels. That means if an ability goes from 40 damage to 50 damage with the second level, it'll increase by +10 damage at every level. So if you've got 100 / 175 as your first two, the scaling would be 100 / 175 / 250 / 325 / 400 damage. Same with Cooldown and Mana Costs.
So this ability, regardless of balance, might cost 50 / 60 / 70 / 80 / 90 mana, have a cooldown of 8 / 7 / 6 / 5/ 4 seconds, and deal 100 / 140 / 180 / 220 / 260 magic damage in human form and 100 / 150 / 200 / 250 / 300 magic damage (and 20 / 25 / 30 / 35 / 40% healing) in Phoenix form.
Moving on to the actual damage...this ability is simply obscene. It's a low-cooldown, relatively cost-less (10% of current hp isn't a heavy price, especially on a character with innate healing capabilities) massive targeted nuke with AoE damage for farming/pushing. The secondary form is a slightly weaker targeted nuke...but this one heals you, giving you decent sustain, and thus better nuking power.
BUT WAIT THERE'S MORE. Our human spells deal +50% damage. So this is a 7-second cooldown ability that hits for, at level 1, 150 + 105% AP damage. Do you see the issue with that?
Highly OP, because the damage is so incredibly high, unavoidable, and quickly reapplied.
W - Enbodiment of Spirits/Return Host
No Cost(5 Second CD)
Human - Does 100/150/200/250/300 Magic damage to nearby enemy champions and transforms, accessing new abilities.
Phoenix - Heals Infurnis for 100/200/250/275/275 and transforms, accessing his regular abilities.
Your human form damage scales well! Huzzah!
Your Phoenix healing doesn't scale well. Less good. Also, all transformation abilities in the game are currently relegated to the character's R button...why did you decide to make this one different?
That said...we've got another powerful close range nuke (although with no scaling...which is odd on an AP character). That's not really a problem, except that it deals amazing base damage, has a very low cooldown, and has no cost. And has its damage increased by 50%, meaning you have a level 3 combo for 300 + 105% AP on a 7 second cooldown. Obscene.
The Phoenix side is simply OP. You have a massive, 0-cost heal on a 10 second cooldown. It doesn't scale, but it's still ridiculous for someone with as much offensive power as Infurnis has.
E - Mind Flame/Rejuvenation
No cost/100 mana
Human - Marks target enemy Champion, dealing 50/100/125/150/300 and 60% of AP to target over 5 seconds and causes Infernis's spells to do 5/6/7/8/10% more damage to target.
Phoenix - Heals target allied champion for 50/100/125/150/300 and 60% of AP to target over 5 seconds and heals Infernis for 25% of the amount over 5 seconds. If Infernis targets himself, he will not heal for an additional 25% but the spell has 50% increased effectiveness.
The damage of the human spell is finally roughly balanced: with the 50% increase this is 75 + 90% AP over 5 seconds, which is solid but not hugely OP. Not sure what the cooldown is though, which is important. The spell damage increase also isn't OP...but it's odd to see it on a character who already has that mechanic on his Human form passive.
Phoenix is, again, probably OP. You have this character with huge burst potential and very low cooldowns, who you try to balance by having him pay a % of his current health (which eventually becomes so tiny it hardly matters)...and then you load him down with incredibly easy-to-access healing. He has THREE abilities that heal him currently: his passive, his W, and now his E. And, since his offensive abilities don't cost mana, he has plenty of time to regen the mana necessary to keep him in lane for a long time.
R - Apocalypse in Flames/Rebirth
50% of Current HP/No cost(Seperate Cooldowns, 170/150/120 in human and 150 in Phoenix)
Human - Does 500/1000/1600 damage and 50% of AP in a large AoE(about the size of Zyra's R). Cannot use if HP is below 80%.
Phoenix - Engulfs Infurnis in flames for 5 seconds, causing all of his spells to heal for an additional 10% and Rejuvenation to deal the same amount of damage healed in magic damage to a nearby enemy champion. In addition, dying while engulfed will revive Infurnis after a 2 second delay in Phoenix form and reset the duration, effectively making Infurnis immortal until the duration is up. Infurnis may not use Return Host throughout the duration of Rebirth.
No, no, no on the human form. Simply way too powerful. The base damage plus ability power alone makes this deal an easy 1800 damage (assuming 400 AP), plus 50% from his passive for 2700 burst damage from this ability alone (which is up there with full-combo single-target rotations from most other AP carries). Then add in the 450 from his W (300 + 50% increase), the 450 + 90% AP from his increased E, and the 375 + 105% from his increased Q--that's 4755 magic damage to a single target, and about 3550 damage to everyone around him NOT specifically targeted by his Q and his E.
By contrast a 600 AP Veigar using Deathfire Grasp on a AD carry and using his full combo will deal about 2750 points of damage to that single target. Not only do you deal more damage than that in incidental damage to things that aren't your primary target, but you also need 200 less AP for that calculation (400 in my Infurnis example vs. 600 + DFG in my Veigar example), have lower cooldowns, have better sustain, have two extra damaging moves I didn't even include (Phoenix Q and Phoenix E with Ult active), and beat out Veigar's burst on a single target by over 2000 damage.
THIS IS A HUGE PROBLEM
The Phoenix Form isn't any better...in fact, it's actually MORE overpowered, since you literally can't die, and that duration can be extended forever. Activate Rebirth and go auto-attack a tower. Die to said tower. Come back to life. Die again. Rinse, Repeat. Forever. The tower will always kill you within those 5 seconds if there are no minions to distract it, so you can literally 100-0 it and move on the the next tower to do the same thing.
It's infinite free lives in the right situation. It makes you immortal in teamfights. It makes you a dueling and split-pushing god. It's simply far to powerful to exist in the game: one revive would be bad enough, but the ability gives you as many as you need, and has additional effects.
CONCLUSION
Infurnis is strictly overpowered in basically every part of his kit. Of his entire kit (all 10 abilities, counting his passive) only his Human form E (Mind Flame) and his Phoenix form passive are anything close to balanced. The entire kit needs a huge revision, and a tighter focus on what Infurnis is and how he should play. Currently he's an single-target burst, AoE burst, self-sustaining, lane-pushing, lane-bully, ability-spamming teamfighting dueling team-healing mage. And that's a problem.
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(specifically the phoenix skin), 
Appearance
A boy slightly younger Annie's age and about her size with bright red hair(bright as in fire red) and phoenix wings.
Lore
Infurnis was not always called as such. A long time ago he was a boy named Adam. Now he is no longer Adam. Now he is merely the host for the spirit of the Phoenix. Growing up in the Voodoo lands, Adam was always fascinated in the stories of a young girl who tamed the legendary monster Tibbers and became a champion of the league. Soon when he became older his fascination turned into a dedication to taming a monster for himself. He had heard stories of a mountain far away inhabited by the tribes of the Rakkor. Inside the core of the mountain was an eternal bird made of flames, waiting to rage and burn, turning the mountain into a volcanic wasteland. Stealing away his mother's teleportation spell books, he teleported into the center of Mount Targon, into the wings of a flaming bird. Surprisingly, he did not feel scared, but comforted by the bird's wings strangely not burning him. The bird spoke to him: "we are as one" and the bird's spirit went into Adam. The Phoenix gave him new powers and fueled him with thirst for more power. Now he is Infurnis. He seeks the Burning Vengeance to prove who will burn in a fight of fire and fire.
