So, I would like to know how to make an amazing alchemist without being an artificer. Cause I don't have Eberron thing. I am thinking about making an alchemy-based subclass. Maybe for wizard, sorcerer or warlock. Warlock sounds best to me. So, any ideas? 

You could still be an artificer/alchemist without being in Eberron.
You could roll one in Forgotten Realm or Dragonlance if you want as long as your backstory makes sense and if your DM is cool with it.
Use the 5e Character App on the Google store to help see what you can roll.

I mean I don't have the book, in real life or on the app. So that is out the question. Any other ideas?

What about warlock or wizard?

You could conceivably take any class, say a rogue (since a thief might be able to do cool stuff with Fast Hands, or Arcane Trickster if you still want the casting flavor), and take the Guild Artisan background so you can grab the Alchemist Tools proficiency. Then you can play v.human for the lv1 feat or just wait till level 4 to take the Healer feat, which gives you proficiency with the Herbalism Kit as well as some bonus healing ability. Ta-daa! You can now craft any potion (including healing potions) using either the Phb crafting rules, Xanathar's rules, or talk to your DM about making homebrew modifications to the existing rules. 

Now, this straight-up-alchemy build is going to rely on having regular downtime, but if your DM is even fairly permissive, you should be able to get some mileage out of it. You'll need a lot of gold, but the plus side here is it gives you something to actually spend your money on, which is often a wall people run into in this game. You're most likely not going to be creating high-tier potions of legend, but if you get to making a good amount of low-to-mid range healing potions, poisons, oil of Slipperiness, alchemist's fire, etc, you'll probably get your money's worth out of the character choice. 

I feel like a subclass devoted to alchemy would be cool. Transmutation wizard seems ok, but would like to have warlock one. Maybe homebrew? Maybe a pact: the alchemist's pact?

I think you're pretty dead set on Warlock my man, might as well go for it. Do whst you wanna play :)

How do you think it would work? Cause I don't know what to call a patron who is an alchemist. First abilities would be able to craft certain potions and other alchemical stuff. Don't know about the others.