Completing Oozing With Character is straightforward, but gets tricky and tedious when you add many players to the mix.
To get credit for "Ooz grown," you use the quest ability Command Ooz on a NEARLY dead mob, make sure you use it on a mob before it is slain, not on the corpse of the mob.
Oozing With Character WoW
Mobs that your Ooz will consume are Shardhide Den Mothers, Shardhide Alphas, and Shardhide Growlers. The Ooz will not consume the gorgers, scavengers, or the Shardhide Cubs.
Keep in mind that any player, whether on your faction or not, can "steal" credit by using their Command Ooz action button on a mob you are whittling down and planning to use for credit. Because of this, try and head to locations in Korthia where the Shardhide mobs are further from the beaten path, lowering the chances that someone will come in and yoink your credit.
Otherwise, growing your Ooz will be a nice way to garner some reputation for Death's Advance, as well as picking up the achievement "Friend of Ooz," earned by completing the Oozing With Character daily ten times.