제가 정말 죽을 죄를 지었습니다 = 제가 (I) 정말 (really) 죽을 죄를 (a capital sin/crime) 지었습니다 (have committed). = I have committed an unforgivable deed. It is a common phrase of admitting guilt (and asking for leniency) said by a criminal under interrogation in the old times (whether it's historically true or not, it has become a stock phrase in dramas). As for the second question, although there are two 을 and 를, most people probably don't even notice it because "죽을 죄" is almost like a single word. When a phrase becomes a common concept like this, it doesn't seem to matter what it has in it, because it hardly even register in the mind of the listener. There are many such phrases: [먹을 것]을 좀 사와 (Go buy some [things to eat]), 일행은 [갈 길]을 재촉하며 서둘렀다 (The party hurried themselves not to fall behind on [their way], [할일]을 다할 때까지 쉬지 않겠다 (I won't rest until I finish [the work to be done].

This is my very limited understanding of the Korean language. The two 을/를 are different. The verb stem in 죽을 죄 is 죽다, it is not an object (it's a verb). 죄, on the other hand, is an object. And the 를 that follows reinforces that. Elaborating on Tony's example, 먹을 것을 좀 사와. It means "you go out, buy something that we can eat (at a later time)". 먹을 것 = things that can be eaten (edible things). Again 먹다, a verb.