Using a weapon with a critical multiplier, such as the hunting rifle or bladed gauntlet, greatly increases your chances of a critical without using an extra perk. A character with 5 Luck, Finesse, and a hunting rifle has a 20% critical chance outside of V.A.T.S.
In Fallout: New Vegas, the Built to Destroytrait increases critical chance by 3% while increasing the rate at which your weapons degrade.
Characters with very low Luck arguably do not benefit as much from Finesse, as they are not typically reliant on critical chance to be effective. This also holds true for characters that almost always attack from stealth - sneak attacks are always criticals and do not rely on critical chance.
In the Official Fallout 3 Strategy Guide, under the Luck section, it states incorrectly that you need a minimum of 6 Luck to acquire the Finesse perk. There is no minimum Luck score (or other prerequisites, for that matter) to select it.
In the Official Fallout: New Vegas Strategy Guide, in the Critical Chance section, it states that Finesse adds an additional 15% critical chance to unarmed and melee attacks instead of the usual 5%. This is not stated elsewhere in New Vegas material.
In Fallout: New Vegas, having the Finesse perk and 10 Luck while using the 1st Recon beret and a weapon with 5x critical chance, you can have a 100% chance of getting a critical hit.