Well, I'd summarise the X-Files as (no spoilers):

• Mulder has his sister (purportedly) kidnapped by aliens as a child.
• As a result, Mulder is an occult nut who wants to spend all his time investigating supernatural stuff.
• For some reason the FBI lets him spend all his time doing that. (Kind of nepotism maybe?)
• They send Scully to go watch him, but she does a terrible job of that.
• Every episode there's a different monster or paranormal thing. Mulder says something like "It's Bigfoot", Scully says "That's silly, Bigfoot isn't real", then they see it is Bigfoot but Scully tries to rationalise it away.
• At some point Mulder gets tangled up in government conspiracies.
• He makes it his side job to try and crack those conspiracies, but continues his monster-of-the-week formula of normal paranormal investigations for the most part.
• As time goes on, there's more and more conspiracy unwinding stuff, but they never fire him or kill him because one of the secret cabal members has a soft spot for him.

More stuff happens in the later seasons, but I'll leave it out for spoilers.

TLDR; Mulder is not anti-government / FBI, he's anti-secrecy and anti-conspiracy. He's an occult nut who just wants to engage in his hobbies on the government dime.