Crit-fishing with a melee build...
1) I think the most important thing is to have a reliable source of getting advantage. The best one imo is reckless attack (spells might not work or counterspelled/dispelled, familiars/mounts can die, abilities that recharge on rests or expend resources might not always be available). The downside with reckless attack is that it makes you more vulnerable, so having additional ways to gain advantage is still important. Yet, reckless attack (hence barbarian 2), is still a top priority imo.
2) After you make sure that having advantage is taken care of, the next thing to do is to find a way to spike your damage when scoring a crit. Sneak attack is a very good way to go about this, as it does not rely on resources (though it requires commiting to a LOT of rogue levels). I think there is a better way to go about this, though. I would do this, via a combination of extra attack (to make the most of the increased crit range and crit damage), GWM (grants bonus action attacks on crits/ or kills, and pairs exceptionaly with triple advantage), and smites (either paladin ones, or warlock ones granted through an invocation -warlock smites are better, and hexblade is also good as it can expand your crit range with some restrictions in place, but the paladin offers some goodies too). Perhaps sneak attack can end up on top after the end of a very long adventuring way, but if you build carefully and plan for a good spell slot progression, then extra attack and smites will have more impact, and a better nova potential (assuming you dont do an awful job handling resources).
3) You have to increase your crit range. This can be done either by going with champion, or with hexblade's curse.
4) Have some defense against effects that poison you, frighten you, restrain you, etc. As if something imposes disadvantage on you, its game over.
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So, all that said, I end up with 2 builds in mind:
Build 1
Halfelf, Barbarian 2/ Hexblade 3/ Bard (Valor) 15
Stats: STR 16, DEX 14, CON 14, INT 8, WIS 12, CHA 14
Feats: Elven accuracy, GWM, plus one more (sentinel, mounted combatant, lucky, str bump, blade mastery, are all some good choices)
Ways to get advantage: Reckless attack (main), faerie fire (secondary)
Spell slots: 4, 3+2, 3, 3, 2, 1, 1, 1 (used with warlock smites or with AoA)
Extra attack from valor bard 6
Increased crit range from hexblade's curse, with restrictions on its applicability.
Main drawback: Due to the split in the classes taken, it will take some time for this build to come online ( and I am not even sure on what the level progression should be). Secondly, this build does not have very good saves, so it's vulnerable against effects that impose disadvantage on us.
Build 2
Race: Halfelf
Character level: 14
Class: Barbarian 2/ Paladin (vengeance) 6/ Bard (Lore) 6
Stats: STR 16, DEX 14, CON 14, INT 8, WIS 9, CHA 16 (or sth like that)
Feats: Elven accuracy, GWM
How to get advantage: Reckless attack, vow of enmity, faerie fire
Maxed slots till 4th level for smitting on crits.
Any higher level spell slots to be used with AoA (grabbed through additional magical secrets) when raging. (Take care so that the spell slots of 5th level or higher to match the number of rages you have per day, so that you can always have AoA on when raging).
Find steed and AoA make for a very strong mount.
Aura of protection for boosting your saves to guard against effects that impose disadvantage (resilient wisdom is a good feat to be picked with your next ASI).
At rounds when you didnt crit or kill something, you can use your bonus action with bardic inspiration. And you can use cutting words with your reaction too. All that even when raging.
If against one strong opponent, you can forgoe reckless attack in favor of VoE. That will probably save you some pain.
The only thing really missing is an improved crit range. We could take 3 levels in champion with our remaining levels, so that we take care of everything missing. But grabbing another or two barbarian levels for a path and an additional rage per day, and perhaps even for another ASI is a good idea too. So is grabbing more bard levels for more and higher level spell slots for better use of AoA (+rage), and for extra spells (for when you are not raging -remember, you have VoE too, so when not using reckless attack you might as well save no your rage and rely on VoE and spells-, or for out of combat use), even for magical secrets at bard 10. Lots of options.
I think I would go with barbarian (wold or totem, depending on group comp) 4/ paladin 6 (vengeance)/ bard 10 (lore), and do without the increased crit range from champion (though that really hurts crit fishing but balances out the build more nicely overall imo). If really focused on crit fishing, I would probably go with barbarian 3 (wolf/totem)/ paladin 6 (vengeance)/ champion 3/ bard 8 (lore).