The “light blade weapon group” is distinct from the “light weapon category” and not all members of the former must be members of the latter, despite what the name would imply. Yes, this is dumb.

So anyway, the rapier is a light blade, but not a light weapon; it is a one-handed weapon. It behaves like a light weapon for the purposes of Weapon Finesse only (and cannot receive 1½ Str to damage if you use two hands with it).

Note, however, that (barring a class feature that says otherwise) you will take large penalties for wielding a one-handed weapon in each hand. Two-weapon fighting requires that one of your weapons (designated “off-hand” during any full-attack in which the two-weapon fighting combat option is invoked) be light to avoid these penalties. Since fighters tend to prefer to stick with a single weapon type because of feats and class features that function only with a single weapon, most two-weapon fighters use dual light weapons. The kukri is the typical 18-20 threat range light weapon. Light weapons can benefit from Power Attack, but not from the extra Power Attack damage you get from using both hands on one weapon (but that wouldn’t work with dual-wielding anyway).

Finally, I just wish to warn you that dual-wielding and critical-hitting builds tend to be on the low end of fighter damage. Two-handed weapons have much greater potential. You should be aware of this going in.

Not with unarmed strikes

As of the first Player's Handbook errata unarmed strikes no longer count as weapons.

Instead of using a weapon to make a melee weapon attack, you can use an un-armed [sic] strike: a punch, kick, head-butt, or similar forceful blow (none of which count as weapons).

The Brute Damage requires a weapon:

Whenever you hit with a weapon that you’re proficient with and deal damage, the weapon's damage increases

This means that Flurry of Blows can not use Brute Damage as it only uses unarmed strikes (emphasis mine):

Immediately after you take the Attack action on your turn, you can spend 1 ki point to make two unarmed strikes as a bonus action.

You could use a Monk Weapon with Martial Arts

Martial Arts affects certain weapons in addition to unarmed strikes (emphasis mine):

• You can use Dexterity instead of Strength for the attack and damage rolls of your unarmed strikes and monk weapons.

• You can roll a d4 in place of the normal damage of your unarmed strike or monk weapon.

If using a monk weapon, the Brute Damage would apply.