lithium grease is too thick of lube for a derailleur, IMO.

sounds like you're having a cable drag problem... a new derailleur should not require lubing.

look for kinks in the cable, sharp bends in the cable casing, burrs at the ends of the casings, and possibly the wrong cable casing type.
Shift cables need STRANDED casing to help limit cable tension being lost to casing compression... the old style coil reinforced casings are not up to the indexing task, and brake cable housing is the wrong interior size, plus is usually a coil type casing.

you DID install new cables, right?

there's a chance you have the cable anchored to the der. in an incorrect position, too... the cable is clamped behind the allen screw, right? if clamped in FRONT of the allen, the pull ratio will be altered...

you didn't lube the CABLES with that grease, did you? if so, start cleaning now.

the Alivio ders. and the ST-E51's SHOULD work fantastic together.... i have a bike here right now set up with them, actually... 3x7 setup, and the customer has an 8 speed cassette on the rear wheel... still shifts fairly close to correctly...

He wants a 3x8 shifter set installed... upgrade! going on today.

Question: just how tight does the CHAIN act when you have the shifting issues? and have you gone through the "B-Screw adjustment" procedure yet? the der. should look about parallel to the chain stay tube...