Maybe Brigand?

Here's a partial summary of what you get from different classes (and don't forget that you can change classes while keeping the abilities you learned from the previous class, except for the innate passives):
* Brigand: Can deal bleed damage with normal attacks (can't be transferred to other classes), gets skills that cause bleed (shadow bleed in one case) and at least one that does increased damage if the target is bleeding.
* Budoka: Gets a passive skill (that is usable in other classes) that makes it so that every attack has a chance to cause bleeding. (This includes things like spells and maces.)
* Spellshaper's fire invocation causes burning in addition to immediate damage.
* There's a Wild Child ability that gives you an attack bonus when anything nearby is bleeding.

Grim Dawn has entire classes and ability trees based on DoT effects.

Neat, thanks. Wishlisted it.

You can try editing save values.

In Elder Scrolls Online every class from the base game has DoT abilities (I haven't used the two new classes from the expansions yet but I think they have some too), and for example most Magicka-based Templars have a lot of AOE DoTs in their rotations - I have at least 5 out of the 10 skills on the skillbar. However I'm not sure how good they are PVP-wise if you're interested in that, or in the top leaderboards in raids, but if you want to just play the game like an offline game or do it completely yourself solo (including all the four-man dungeons), all classes are viable for that, and Templar with its healing skill tree is one of the best solo classes.

If you're interested, you can checkout the abilities on the wiki below.

Each class has three unique skill lines, in addition to about 6+ general skill lines for everyone, not counting the crafting skills, and every skill that's an active skill allows you to select a morph after being levelled to the max. So for example after levelling up skill X, you have the option of turning it into morph A or morph B, and each one adds an additional bonus effect(s) to the original skills or sometimes outright completely changes them, but it's rare.