First, does the Keen weapon trait exist in D&D 5e? And if it does, is it possible to make a keen weapon using the fabricate spell?
The caster needs proficiency in smith's tools to make a weapon, I know, but if that is gotten by being a dwarf, or through a background, I was thinking of saying that I made a blade that had a monomolecular edge, which should be sharp enough to be keen.
Keen doesn't exist yet... But no.
To start with, Keen weapons (that receive a bonus) have historically been magic items. They aren't just sharp, they're magically sharp. And magic items are explicitly excluded by the spell:
Creatures or magic items can't be created or transmuted by this spell.
Even if your DM was generous, and ruled that a monomolecular blade would count as Keen without being magical, the spell will still likely fail as worded for another reason:
The quality of objects made by the spell is commensurate with the quality of the raw materials.
I'm pretty sure that steel doesn't take or keep a monomolecular edge for any meaningful length of time. You'd need to find something that does, which is likely to be difficult to acquire in a pseudo-medieval setting.
All that being said... Be sure to ask your DM. The rules say "no," but your DM is free to override them if he thinks the results will be interesting.
The spell Fabricate could not add Keen to a weapon, even if it does exist.
Here's the relevant quote, from the Fabricate spell entry on page 239 of the PHB:
...magic items can't be created or transmuted by this spell
Also relevant is this, from the same entry:
The quality of objects made by the spell is commensurate with the quality of the raw materials.
So while you could create a weapon (if you had proficiency with the appropriate artisan's tools), the weapon couldn't possess any magical properties.
There is currently no Keen weapon property. The only published material we have that includes magical weapons is the Starter Set adventure, the HotDQ campaign, and the DM Basic Rules. None of these include weapons with expanded critical ranges. If there is a Keen weapon property in 5e, it will almost certainly be coming in the DMG.
We don't know whether it is possible for a PC to craft magical items yet either, but Fabricate can't make magical items (credit goes to the other answers). Crafting rules are another thing that should be in the DMG.
It's probably possible to create a sword with a monomolecular edge, using the right materials, it just wouldn't hold the edge beyond the first hit. In my Interpretation, keen weapons possess monomolecular edges, enchanted with localized unbreakability along their edges, keeping them from deforming or chipping.
