Olive oil will kill lice, and probably condition the hair at the same time. It suffocates them.
Vinegar will loosen the nits from the hair so you can comb them out better.
I would start with vinegar (and lice comb) because it will also clean the hair. Then apply olive oil and leave it on under a shower cap overnight. Then use the lice comb again before washing the oil from the hair.
One source I read said to use olive oil on on days 1, 2, 5, 9, 13, 17 & 21 to best disrupt the life cycle. This source also did not mention vinegar. Other sources do not mention repeating the olive oil. Other sources say you have to repeat treatment even if you are using a chemical pesticide, due to the lice life cycle.
You can use cheap, low-quality olive oil, or use a completely different oil.
I have read that Listerine could be more effective than vinegar and might not require olive oil.
Cetaphil is highly regarded too.
Here is a mange cure that might work for lice: Mix 50% apple cider vinegar and 50% Borax (regular 20 Mule Team Borax from the laundry soap aisle). Soak the hair and scalp for 10-30 minutes, then rinse.
This is the website of a Stanford-trained dermatologist who recommends using only Cetaphil cleaser.
Instructions:

I caught lice twice as a child. The first time, from an animal (my entire family was contaminated. It was a nightmare)
The second time it was the normal lice from school.
My mother coated my head with mayo before bed. Shower cap. Then the nitpicking for an hour every morning - followed by a wash to get out the mayo. Repeat every damn night for a freaking month.
Frustrating for everyone.
Sounds like a recipe for Italian dressing :-)