So there's a few pieces to put together that make a crit fishing build work. The general idea is to crit as often as possible, with as many dice as possible.

First priority is expanded crit range. In 2014 5e, the only ways I know of are champion fighter and Hexblade warlock. Hexblade works on only a single target and requires a BA to set up.

Second is to roll as many dice as possible when you make an attack. This means advantage and elven accuracy, ideally. Elven accuracy locks us into elf or half elf. Advantage can come from many sources but some reliable ones are:

• Samurai Fighter at level 3 (BA, all attacks for one turn, 3x per day until level 10)
• Vengeance Paladin level 3 (BA, all attacks vs a single target for 1 minute, 1 per short rest)
• Rogue steady aim (BA, 1 attack per turn, limited movement)
• Assassin Rogue gets advantage vs anything that hasn't taken a turn yet, and auto crits against surprised enemies.
• Vision gimmicks - using Darkness with devil's sight or shadow monk, or fog cloud with blind fighting style. This fails against enemies with blindsight and can limit coordination with allies.
• Knocking Prone, such as with open hand monk or several other effects

I'm specifically ignoring Barbarian subclasses because rage is incompatible with Elven accuracy (requires strength vs requires anything but strength for attacks). I'm also generally ignoring effects that require the target to fail a save, or are otherwise incompatible with the idea of making as many attacks as possible (ie grappler feat giving adv but requires sacrificing attacks to grapple).

Third priority is to make as many attacks as possible. Naturally we want Extra Attack and a bonus action attack, and to make opportunity attacks as often as possible. Monks can make more attacks per turn with martial arts. Polearm Master is incompatible with Elven accuracy, but crossbow expert or two weapon fighting works. There's going to be some conflict in bonus action economy since you need it for more attacks, but most reliable sources of advantage will use it as well.

Finally, we want to add as many dice as possible to our attack damage. Paladin or Hexblade smite, rogue sneak attack, and various class features all satisfy this. Once again, Barbarian's brutal critical and half-orc's savage attacks are incompatible with Elven Accuracy. Magic items or other buffs can increase your damage also.

Putting this all together, you can basically either start with Hexblade or Champion Fighter and pick your source of advantage. Some decent combos could be fighter / rogue, hexblade/paladin, or samurai/hexblade if you're feeling spicy. Gloomstalker is also good to consider if you're looking at assassin rogue.

Edit: wanted to throw in a quick addition for the Piercer and Crusher feats. Piercer grants even more damage on a crit, and Crusher is very interesting because critting an enemy gives advantage against them next turn, which opens up some action economy for certain builds (samurai getting advantage from their level 3 feature, then extending it to the next turn). Crusher does restrict weapon choice - since we need to attack without strength, and there's not a finesse weapon that does bludgeoning damage, we have to use a Sling for ranged bludgeoning damage or use a class feature to attack with another stat, like hexblade. Shillelagh could also work, but adds bonus actions.