Despite being Avshar's left hand, Alhazred probably knows to take orders from Ulquiorra as if he was Avshar in the event that the latter's missing. He'll largely be okay with this, but will probably find Ulquiorra a little offputting. He likes despair and futility as much as the next guy does, but it gets to be a little depressing sometimes. Al's a demon that loves life and all.

Overall they probably share strategies and execute them as necessary. It might be a difficult relationship since the two probably reserve a level of respect only for Avshar. But a working one nevertheless.