The DMG pp. 257-258 has a bit of what you're looking for: descriptions of fourteen sample poisons, with damages and costs. A few individual descriptions contain enough detail to make (some) ingredients obvious--see Carrion Crawler Mucus and Serpent Venom, for example.
As for crafting poison...
During downtime between adventures a character can use the crafting rules in the Player's Handbook to create basic poison if the character has proficiency with a poisoner's kit. At your discretion the character can create other kinds of poison. Not all poison ingredients are available for purchase, and tracking down certain ingredients might form the basis of an entire adventure. (DMG p.258, emphasis mine)
Even combined with the PHB Poisoner's Kit (p.154: "Proficiency with this kit lets you add your proficiency bonus to any ability checks you make to craft or use poisons.") and Crafting (p.187) sections, there's not much here to go on. Combining all three sources could go something like this:
- Character would like to make Assassin's Blood poison (DMG 257-8). It costs 150 gp to purchase.
- Crafting such a poison will take 30 days (1 day per 5 gp) along with proficiency in the poisoner's kit (PHB 187).
- At the end of the 30 days character will make a check to see if the crafting was successful. Let's base this check off of the example of Serpent's Venom on DMG p.258:
- It's a DC 20 Nature check to harvest the serpent's venom, which creates a poison with a DC 11 Con save. Since the Con save for Assassin's Blood is only 10, your crafting DC for the Assassin's Blood poison should be less than 20.
- Failing the harvest-check badly can go badly for the character. Ditto brewing Assassin's Blood (or distilling, or decanting, or whatever).
I think that's all the rules have got for you.
And here's my bonus inspiration for GMs that read this far: the bottle of 'wine' I saw at my doctor's office this afternoon...
Yes, that's a bottle of wine with an entire cobra in it. And yes, in the cobra's mouth is a whole scorpion.