Let's break this sentence down into its simplest part.
権利【けんり】があるのか?
right [SUBJ] exists [EXPLANATORY] [QUESTION]?
Is there the right [to do something]?
Note that ある can also be translated as "to have", if this verb is used in relation to a person. The person "having" the thing is frequently marked with the topic particle は, as in 私【わたし】[は]{●}自転車【じてんしゃ】がある → "as for me, there is a bicycle / a bicycle exists" → "I have a bicycle".
The person can also be marked with the dative / locative particle に. Translating the construction [PERSON]
に[THING]
がある might be a bit like "by [PERSON], there exists [THING]". The phrasing is odd from an English-language point of view, but it isn't that uncommon in languages around the world -- we see close structural analogs in such far-flung languages as Hungarian, Navajo, and German. Even English, in certain cases.
Adding the person in, we get:
僕【ぼく】に権利【けんり】があるのか?
I [LOC] right [SUBJ] exists [EXPLANATORY] [QUESTION]?
Is there the right [to do something] by me / in me?
Do I have the right [to do something]?
And lastly, as I suspect you're already aware, the bit in the middle tells us specifically what kind of 権利【けんり】 or "right" we're talking about here -- the right to 勝手【かって】に人【ひと】を裁く【さばく】, or "freely judge people".