Like his boss Robin Hood, the friar of the outlaws goes by many names. When we first meet him in the forest, he introduces himself to King Richard I (himself in disguise as the Black Knight) as the "Holy Clerk of Copmanhurst." He likes to live large, and his position as a holy man certainly isn't going to stop him. He is extremely cagey about admitting that he drinks a ton and frequently eats deer that supposedly belongs to the local Norman elites. But he still likes to pretend that he really is a religious man living in solitude and praying in the woods.
As King Richard I slowly convinces the friar of his good humor and relaxed approach to poaching and drunkenness, we gradually figure out that the friar's "holiness" mainly consists of swearing frequently to English Saint Dunstan. Otherwise, he likes a good fight as much as the next man, and he frequently puts on the green clothes of Robin Hood's men to rob people passing through the forest. Posing as a hermit and a monk provides a cover for Tuck's less-than-holy activities, since he has an acceptable reason to spend all of his time in the forest. As soon as Robin Hood knocks on his door to demand that he join their siege of Torquilstone, though, Friar Tuck enthusiastically joins in.
Friar Tuck mostly functions as comic relief in the book. His drinking songs with King Richard I and his drunken fight with Athelstane as Athelstane is escaping from the Priory of Saint Botolph are supposed to make us laugh. It's a really old-fashioned kind of humor, since comedy based on the hypocrisy of a priest who is drunk all the time might seem less amusing or acceptable to modern audiences. But the drunken friar thing is a big theme in medieval literature, so by including his own version of Friar Tuck in Ivanhoe, Scott continues to imitate actual medieval literature in his 19th century novel.