I've done something similar in the past without actually making it a build, mostly used skooma to have some fun and maybe lay down some quick damage.
However...........I am bored.
I decided to make a build revolving around chugging large amounts of skooma and just slamming down the hurt. Also working around the drained agility and intelligence to an extent.
This isn't a guide really since im still working it out, so tips would be much appreciated.

Race: Wood Elf (because they are my homies and for this build i'm not caring much about the racial bonus)
Birthsign: The Warrior (boost in strength and endurance could be handy although thief might be nice too)
Specialization: Combat
Favored Attributes: Strength and Endurance (or strength and speed)
Major Skills: Hand to Hand, Block, Athletics, Heavy Armor (for boosting endurance / health), Alchemy, Sneak
Last perk is a toss up between Acrobatics and Illusion. Illusion could work well for the frenzy and paralyze spells, but whether or not you can cast this with a gallon of skooma in your belly i'm not sure.

If you see something you would change please let me know! Still open for ideas on this build!

Side note I will probably be using a mix of heavy armor and normal clothing.

I might consider the Atronach sign. Skooma's effect on Agility is temporary and wears off in 20 seconds, but its effect on intelligence is permanent, until restored. You could always play a character that uses no magic since his intelligence is so damaged by skooma, and just rely on resting and alchemy for healing. The 50% spell absorbtion would come in handy just to reduce damage taken from spells, and the stunted magicka would be of no consequence to a character with zero intelligence. Plus, the 200 magicka bonus might actually allow the character to use magic even with a fully damaged intelligence that was reduced to zero.

I don't know what will happen if your intelligence is further damaged by more skooma after it is reduced to zero. Will it go into negative numbers and start eroding your 200 magicka from the Atronach sign. Could be interesting to find out.

Hmm actually that might work pretty well. I've never seen skooma dip me into negative numbers unless it's an effect that doesn't show up in your stats, which is possible.
Thanks for the input, i'm gonna work with it and see how it does.