Was Grimiore Heart aware that Zeref was still alive?
Ultear controlled Jellal to build the R-System, so the Magic Council focuses on him and not on the Grimoire Heart, who could then move in the shadows to continue their search for Zeref. (Tower of Heaven Arc)

So to answer your question, Yes, Grimoire was aware Zeref was alive and used Jellal as a decoy to search for him. After finding him launching an attack on Tenrou Island (Tenrou Arc)