I know those are 2 completely unrelated periods of time, but they're both legal systems where the judiciary explicitly or implicitly favored the powerful: the Romans ascribed more weight to the assertions of the wealthy than the poor, and while an Icelandic court might be able to levy small fines and declare outlawry on a guilty party, it was up the aggrieved to actually extract "justice"/revenge (along his allies, benefactors, friends or family, which the wealthy probably had in abundance and of which the poor had few). And that's not even getting into whether or not the aggrieved will be able to "get his (usually his) day in court".
That is an interesting question. I can, however, only offer part of an answer. During the Roman Republic and decreasing in importance during the later Imperial Period this part of Roman cultural life depended on the system of Patronage. This can be understood as a form of social contract in which a patronus, who was an affluent member of the Roman nobility, amassed a following of clientes.
While the patron offered financial and legal support to his poorer and considerably less aristocratic entourage, the clientes could be mobilized to support the election campaign of their patrician or vote for legislature put forward by him. Another peculiar aspect of this exchange is the salutatio. In this daily 'morning ritual' the entirety of a patron's clients would come to his residence and greet him. Especially during the last century of the Republic this would gain importance as one way among many in which patrician families would compete with one another. Therefore, it was in the interest of a patron to garner as much support as possible, since the longer the queue in front of your urban villa, the more you celebrated a triumph etc., the bigger your *cough* prestige was. Finally, freed slaves or liberti were automatically added to the number of clients a patronus had, so he was consequently also responsible for their legal protection, which might answer parts of the follow-up question by Einkill.
Now back to the question at hand: Since it was common for a young patrician to be educated as an orator he could defend his incriminated cliens in court if need be or even find someone to do it for him. Regarding non-citizens, we have an interesting example in Cicero's oration pro Deiotaro in 45 BCE. Here Cicero takes up the defense of a Tetrarch from Galatia who was accused of attempting to assassinate Caesar by his grandson and his personal physician. Curiously, this litigation is carried out with Caesar as the aggrieved party, the judge and owner of the home in which this all takes place. Cicero of course saves the day in case you were wondering
To conclude, while the judiciary system of Rome probably has been tipped in favor of the nobility anyway, this social contract of patronage was one mutually beneficial way to at least mitigate the power dynamic between aristocracy and plebs. Hope this helps to answer your questions.
Goldbeck, Fabian (2010). Salutationes: Die Morgenbegrüßungen in Rom in der Republik und frühen Kaiserzeit. Berlin: Akad.-Verlag.
Gotoff, Harold C. (1993). Cicero's Caesarian Speeches. Chapel Hill: U of NC Press.
Nicols, John (2013). Civic Patronage in the Roman Empire. Leiden: Brill.
Wallace-Hadrill, Andrew ed. (1990) Patronage in Ancient Society. London: Routledge.
As a follow-up to this question, the New Testament talks about petitioning Caesar as the right of a Roman citizen. Could any Roman citizen just demand a judgement by Caesar, or was this more limited in accessibility?