I'm playing a Rouge assassin with my current party and I had some questions come up with my DM about poisons. 

My character has a backstory where he knows an insane amount about plants since he grew up sustenance hunting in the woods, and was taken in by a healer woman. In my inventory I've written down a list-approved by my DM- of both medicinal and poisonous herbs and plants he would have access to. Part of the story is that I take care of another player's father by treating him with homemade cough syrup. I know that in the world you can buy poisons, but with his backstory ability and with the limited supplies in his inventory I think that I should be able to make my own poisons with a time penalty. If I run out of supplies, then I will have to forage for more. My DM is not super happy with my suggestion that I could make my own poisons, and I understand the reasons why. So, I'm looking to balance it- or possibly get to a higher level (currently 6) so I can do this. (it would also help if i could make potions to heal myself as well, but there are others in the party that can heal.)

Of course, ultimately the decision falls to my DM, and I will respect it, but I feel like some sort of balance can be had. I've also been looking through manuals about character creation to see if this ever becomes relevant or a skill later on, and I've found nothing about it so far except in a super old 1979 DM manual. But that also says assassins get a 50/50 chance of insta-kill so... 

I want to do right by my character, but I also want to respect my DM and what balancing mechanics she has in place. So ultimately, my question is: Is there a poison/poison making proficiency I can get naturally within the game, or are there any suggestions on how to balance an ability like this? 

The poisoner feat in Tasha’s might be a good place to start. 

There is a tool proficiency, the poisoner's kit. A Rogue with the Assassin subclass gets proficiency with that at level 3. You could also make a Custom Background that gets a tool proficiency and choose poisoner's kit for that.

There is also a Poisoner feat from Tasha's Cauldron of Everything, which does more of what you probably want. That'll depend on if and how you can take a feat. That'll depend on your DM, but these are from official materials.

My character has a backstory where he knows an insane amount about plants since he grew up sustenance hunting in the woods, and was taken in by a healer woman.

This is covered entirely by proficiencies. In this case, poisoner's kit. You can't just make stuff up in your backstory to justify extra perks. A 500 year old elf character could feasibly know like 15 languages and be an expert in arcana, history, performance, etc. but they don't get any more starting proficiencies than anyone else.

NOTE that from memory, the rules in the DMG around creating/crafting poisons, tend to be with regards to gathering them from creatures rather than plants.  If you take the herbalism kit and have some skill around that you might be able to argue with the DM that you can gather relevant plants/herbs and create poisons from them as well.  In general, poisoning is considered very low for the impact it can have.  I'd suggest talking with your DM about the ability to create more powerful poisons that either have a higher DC or cause more damage.  You might even be able to argue further flavour with regards to poisons that cause conditions such as paralysis rather than damage and even delayed damage rather than immediate poisoning.