Alright, let's get started with this review. I feel this champion is very interesting and unique. He has some good spell concepts. Onto the spells:
Passive: Venomous Strike (Passive) (On Hit) (On Hit) (Target Selected) (1)
His strikes deal poison damage over a period of 3 seconds (+.1 second per level) that deals damages 15 (+1 every level) health points every second and stacks up to 10 times, in damage.
Opinion: I think you were going for a spell like Teemo's Toxic Shot. The only problem I really have, is that this is a passive. To me, this reads, "every hit you're poisoned." So unless you can get away from this Hydra for 3 seconds, you will always be poisoned. Although the damage is low, this could still be nerfed to say, "has a chance to poison" or "only poisons every 5 seconds.". I think I got what you meant, but since it is a passive, it will need to be a bit nerfed.
Q Ability: Thrashing heads (Active) (Area of Effect around user) (10/12.5/15/17.5/20) (300 units in full circle around) (0)
His heads flail around him in a circle, dealing 100/150/200/250/300 per unit physical damage.
Opinion: I think this would be very cool to see in-game, however, all it does is damage. If we could add some things here, for example, Knockback, this ability would be more fun and usable. You have the damage correct, and everything else is fine, but if we add knockback to this ability, it will make it a much more fun spell to use. Overall, a solid spell.
W Ability: Venomous Spit (Active) (Target) (8/9/10/11/12) (850 units) (10)
He spits from his main head to deal 50/75/100/125/150 (+.4 per physical damage point) physical damage then for 3/4/5/6/7 second(s) for 10/15/20/25/30 (+.3 per magic point) magical damage.
Opinion: Your average DoT, nothing fancy here. The range is a bit too short however. Maybe we can bump that up to maybe 1000? That would perfect the spell. Overall, this spell is not too shabby.
E Ability: Snapping Bite (Active) (Target) (10/13/16/19/22) (400 units) (5)
He bites at his enemy, dealing 150/225/300/375/450 (+.2 per physical damage point) and stunning him for 1/1.5/2/2.5/3 seconds.
Opinion: Your state of the art stun. However, the range should be closer to melee range, about 300? The damage is also very high, maybe shorten it down, while buffing the AD ratio? (+.2 to +.6). Also the stun is a bit to long, maybe make it go from 0.5 > 2.5. No one likes a 3 second stun. Overall, this spell only needs some number fixes.
R Ability: Regrowth (Active) (Self) (90/90/90 after deactivating) (15)
Arrick regrows 2 heads (+1/(30/28/25/23/20) of his total hp) for every 1/(10/15/20/25/30th) of the maximum health missing, and increases his attack damage by 25 for each head regrown. This lasts for 10 seconds before deactivating
Opinion: A bit like Mundo's ult. Regaining health would be essential to this champion however, for he has no way of escaping. This is definitely a good spell and I put my stamp of approval on this spell.
Overall: Your champion is actually quite boring. The skill set is mainly from other champions, but I could live with that. Your character however, stuck me as boring. The lore was well written, but the concept of a hydra did not impress me. I don't mean to be harsh, just trying to give a true review. I think you may want to look into recreating a character, while keeping the same skillset. What I mean is, trash the Hyrdra idea, while keeping all the spells for another character. The character is very important. I mean, we couldn't have a Bunny be using Renekton's skillset, now could we? It would be unorthodox and just plain weird. I encourage you to keep creating champion concepts, but be more keen on the character part of the concept. Thank you.
Also, could you judge my new concept? Do it as harshly as you like, but try to keep it constructive.
Tenelus, the Shadowed Seraph