NPCs don't have classes
Only PCs have classes. NPCs have a statblock that fully encompasses everything they are capable of. They may have features in common with PC classes, but they may have extra features a PC cannot replicate, or be lacking features that a similar PC possesses.
Referring to the Basic Rules for 5E (replicated in the Monster Manual) we get the following statement:
A monster's statistics, sometimes referred to as its stat block, provide the essential information that you need to run the monster.
That's it...nice and simple. There's no extra stuff you have to look up, no progression charts to refer to. You just look at their statblock, and it contains everything they are capable of.
(I get where the confusion may come from...3.5E had 'NPC classes' like the Aristocrat, Expert, Warrior, etc. No such thing is in 5E)