I am a very very BIG Harry Dresden fan from the Dresden Files.
I really want to play a concept where I am a spellcaster who keeps a skull that contains an entity that teaches my character lore and different spells as he levels up. I'm trying to mirror the concept from Harry and I know he is called a Wizard, but he seems like he could be more of a Sorcerer than a Wizard, or even a Warlock.
I want him to have some quirky utility spells along with being a bit of a blaster.
I think in the Dresdenverse, Wizards are able to do many things while Sorcerers and Warlocks can only do a limited number of things but do them really really well. I guess you would call Harry an evoker since he has a knack at blowing things up and using fire. I was also thinking of maybe a Feypact Warlock seeing as his mother Margaret had heavy dealings with the Fey and Harry has a Fairy Godmother.
Not really sure how I could go about this.
I'm assuming you're more interested in modeling Harry's capabilities than in modeling Harry's narrative. To model the narrative, other systems work better, such as the Dresden Files RPG (shocking!

)
First off, as you say, class names are meaningless, as we're transporting between milieus. We need to focus on what the character can DO. In the Dresdenverse, there's two types of magic, Thaumaturgy and Evocation. Thaumaturgy (like making potions, putting defensive enchantments on clothes, wards, and scrying spells) can easily be handled by Ritual Casting.
In terms of his combat magic, Evocation, you want to be able to model
Fuego (some combination of Fire Bolt cantrip and Scorching Ray),
Forzare (Eldritch Blast cantrip),
Ventas Servitas (Gust of Wind), and some defensive shields (Mage Armor and Shield seem the most appropriate). Other spells that do fire, air, or force effects will keep up the feel, since Harry rarely does water or earth magic.
Since Harry uses the same spells over and over, Sorcerer and Warlock are a much better fit that any of the other casters. Plus, Charisma is definitely Harry's best stat, with Con second and Inteliigence tertiary at best.
(Digression: The fact that Harry pisses people off is a terrible argument that he has a low Charisma. He's
trying to piss people off. Succeeding at getting people to feel what you want them to feel is a high Charisma. Add in the fact that he has multiple allies willing to risk their lives for him, and that his magic power is always described as being driven by his will and his high inborn potential, and Charisma becomes his obvious focus for a casting stat.)
So, I'd say Warlock or multiclass Warlock/Sorcerer. Make him a tomelock, and make Bob the tome, and I think you have a very Dresden-ish character.