You can see that the class DC starts as Trained under the Initial Proficiencies section for the class.
At higher levels, class features increase the proficiency in Class DC to Expert and Master, e.g. Rogue Expertise and Master Strike for Rogues, or Fighter Expertise and Versatile Legend for Fighters.
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All proficiency bonuses work exactly the same way.
No bonus if you're not proficient, 2 + your level if you're Trained, 4 + your level if you're an Expert, 6 + your level if you're a Master, and 8 + your level if you're Legendary.
Most classes start out Trained in their class DC, but spellcasters get Trained in spells and spell DC instead. Classes that don't get class DC proficiency also won't have any features or feats that call for it (though it can interact awkwardly with some other feats and archetypes).
Like all proficiency, class DC will increase with your level. It also gets periodic bumps from your class features as you level up like most things.
Important to note that Casters don't get a Class DC. Champion gets both a Spell DC and Class DC by default, and a few martial classes can get a Spell DC from feat choices as well. Casters need to take a martial multiclass dedication to get one.
Also important to note that Cleric is the only class (off the top of my head) that can use their Spell DC as Class DC for crit spec. Spell DC and Class DC are not otherwise interchangeable. I guess Relics and a couple other items, but they specifically say when you can do that.
Also important to note it's difficult for a full caster to increase the Class DC they do get from an MCD. So lots of cool abilities they might want to pick up from a martial class won't work well since they'll remain at Trained forever. Meanwhile, martials can advance their Spellcasting DCs pretty well with a series of feats (or as mentioned, within their own class, essentially by picking up a Focus Spell).
I get that they work the same, Ijust don't know where to find the class proficiency and when it actually increases to master and beyond.
If the class doesn't say that it gets a Class DC proficiency, then it doesn't. If it doesn't say that it increases, then it doesn't.
If you don't have a proficiency rank listed, then your proficiency rank is 'Untrained'. You could still use your Class DC, I guess. But it wouldn't be very good.
Not every class gains a Class DC. Notably, spellcasting classes don't generally get those.
For example:
Rogue:
In initial proficiencies it lists "Trained in Rogue class DC".
At level 11, Rogue Expertise increases the proficiency to Expert.
At level 19, Master Strike increases the proficiency to Master.
Wizard:
In initial proficiencies it does not list any proficiency rank for Wizard DC.
And none of the class features at higher level change that.
Instead, Wizard gets a spellcasting attack and DC proficiency for the Arcane tradition. Which starts at Trained and eventually increases to Legendary through class features. But they remain Untrained in Wizard Class DC.