The Solliloquy is (one of my absolutely personal favorite poems) not as much a "dramatic monologue" thought it could be considered that since that is Browning's style. I would just stick to it being considered a solliloquy whose main idea is to stream the inner thoughts of the main character and the depiction or focus on one other character as a result of the main character's analysis.
In this totally hillarious solo, we have two monks: the main character, who is angry, jealous and vindictive, and his focal victim, brother Lawrence.
According to our main character's rant, Brother Lawrence is basically a hypocrite and a manwhore who eats too much and indulges in just about everything. Interestingly, though, our monk main character describes TO THE LAST DETAIL all the things brother Lawrence does, almost as if he is delighted to talk about these things.
The premise of the whole thing is that the narrator is indeed pointing one finger at poor brother Lawrence while the other three fingers point at himself. The fact that he details brother Lawrence's lechery, lust, and gluttory makes you wonder why in the world a righteous man would just let it go, PLUS why would he stand and watch. That is because the narrator is also involved in these treacheries of the church but, apparently, brother Lawrence has gotten a thicker fare from his doings and enjoys them more. Hence, our narrator is jealous.
Remember that Browning wrote this solliloquy during the most prudish of Victorian periods where religious fanaticism took the place of the inner sanctum of debauchery taking place in the Church, State, and even homes of those who claimed to practice it. This is a clear rebel yell against religious institutions and it takes a clear dig at what Victorians considered moral and righteous.