I want to make a demonologist type character, who has extensive knowledge of fiends and uses them as well as dark magic in battle.  Does anyone have any suggestions on how to make this type of character? (We begin at level 3)

Pretty sure Fiend Warlock is the best way to go for this. Get the Pact of the Chain boon. You might need Xanathar's Guide to Everything to get the right spells to get the right feel.

The Critical Role Blood Hunter would actually work veeeeery nicely as a Demonologist. Profane Soul would probably work best, but Ghostslayer would do well too.

Wizard or sorcerer also work. I your character concept is basically a background (probably sage or cloistered scholar) and access to spells.

I think fiend warlock with pact of tome or chain (quasit) is most appropriate.

I recommend as the guy above me to multiclass Wizard and Sorcerer. Sorcerer allows you to take the Shadow bloodline which is off a fell bloodline. 

A warlock is basically a dude who made a pack with a demon so that could fit. 

If you do not want to have anything to do with demons but just have knowlegde, either a wizard or (This might sound weird Monk) both have reasons to have the knowlegde around demons and can still have abilties demons can have the same. 

Background haunted one seems like a good jumping off point