The Vault Hunter's Relic only drops the chances of white item drops by 5% and then splits that 5% between the green and blue drops. so say if the odds of drops are:
No drop:40%, white:30%, green:15%, blue:10%, purple:5%, orange:Only on named & bosses (and often about 1% or less there) OR 3 borderlands symbols on a slot machine (about a 1 in 5,000 chance)
The Vault Hunters Relic would only make that:
No drop:40%, white:25%, green:18%, blue:12%, purple:5%, orange:Only on named & bosses (and often about 1% or less there) OR 3 borderlands symbols on a slot machine (about a 1 in 5,000 chance)
assuming those numbers were accurate (the orange is pretty close but the others I am mearly speculating) then even if it were modded to be a 30% better chance (thus eliminating white drops all together) it would still only be about: None:40%, White:0%, Green:38%, Blue:17%, Purple:5%, Orange:special.
No modding of %'s will make the Vault Hunters Relic really worth wearing once others become an option. Better to wear a relic that works to improve your strengths (such as an elemental boost on a siren or a brand-specific relic that helps boost your best weapon) or compensate for a weakness (+shield capacity or max health if you find yourself dying too easily).
Best Answer
No, the player who kills it is unimportant. The relic affects global drop values so in theory even enemies not touched by the player (E.g. thresher's killing varkids in the caustic caverns) should have a slightly better drop chances. It is also worth noting that the Vault Hunter's relic only increases the spawn rate of greens and blues while decreasing the spawn rates of white items and even then, the change is miniscule. The relic is also unfortunately the only thing in game that can change these values (at the moment, anyway. Future DLC may change that). You really are better off equipping something else.