first of all, he didn?t knew that sasuke killed Zabuza, he just wanted sasuke to tell him were the sword was.He follows sasuke since he aims to kill his brother Itachi, and his partner is Kisame, and he owns one of the seven swords of the mist.the link is that suigetsu wants to collect all of the swords of the seven swordsmen of the hidden mist and he knew that sasukes team killed sasuke so he told him to bring where he died so that he could get his sword