Yes, there's a way to increase ability scores with a class ability
At 20th level, a Barbarian gains +4 to strength and +4 to constitution as their capstone ability. This meets your requirement of increasing ability scores without items, but is going to prove unlikely to help with increasing a Warlocks damage from lifedrinker.
There is the option of casting the 9th level spell Shapechange, which allows you to change your strength, dexterity and constitution, but once again, this won't help you with your specific problem, because you keep your own charisma. It's also a 9th level Wizard / Druid spell, so it seems unlikely you'll have access to it as a Warlock.
For that one, you're most likely out of luck. D&D 5E has done away with almost all effects that increase or decrease ability scores, to streamline the game and to make it so you won't have to recalculate all your skill and attack bonuses on the fly.
There are still a few monsters that can drain ability scores (the Intellect Devourer and Shadow come to mind), but as far as I can find, there are no official spells or abilities that increase ability scores asides from the Barbarian capstone. Those kinds of things have all been replaced with the advantage mechanic.
On a pure technicality level, Ability Score Increases you get at certain levels are also technically a class feature, so those are also an option, as pointed out by @DaleM, but I assumed you were already aware of those.