The martial classes shine here and most do their damage in different ways, so each are good choices in various situations.

All things equal, the rogue is the best damage in the game. The rogue specializes in dealing massive damage in a single attack once per turn. The Help action grants advantage on a single attack and the rogue is the best class to take advantage of this. They can also stack damage cantrips like Booming Blade with Sneak Attack (again, not limited by the number of attacks per turn). A Battle Master allows them to Sneak Attack twice in one turn, the same goes for an Opportunity Attack and the Sentinel feat.

If the party doesn't have a way of supporting the Rogue, the Fighter is best stand-alone damage dealer. Since they scale off of the number of attacks they have, they benefit way more than other classes from effects that increase damage per attack (Hex and Faerie Fire are great if you can figure out how to get them with a Fighter). The Fighter relies on magic weapons, like Flame Tongue, to scale their damage.

Paladins have the best burst damage but they burn out rather quickly; once the run out of spell slots, their damage goes down signifcantly.

Barbarians have one of the best ways to generate advantage with Reckless Attacks (along with the Rogue's optional Steady Aim).

Monks have a niche here, too. They get the most attacks at level 5 (four attacks) and in a low magic setting, Ki-Empowered Strikes goes a long ways to overcome damage resistances and immunities.

The Baseline: Eldritch Blast Warlock

A warlock with Agonizing Blast using Eldritch Blast should be the bare minimum for damage dealing. This is probably the most obvious damage dealing build in the game, so if we can't do better than this, we should just use this instead.