Vikings definitely had some judicial system. By the time things got written down, in Scandinavia around the 13th century or so, there are lots of references to how things 'used to be done'. Their phrasing also suggests that these laws were older, and were transferred orally (through speech) rather than in writing, so they are a good source for Viking-period law as well. A bit further south, the situation is similar, but laws here were written down in the 8th and 9th century already.

Now, the thing seems to be that in Germanic society (including for the Vikings), laws were valid for a people. This basically means that if you are a Jutlander (native), the laws of Jutland would apply to you, even if you were currently in Sweden. This means that you are, as a foreigner, outside of the protection of the law if you are travelling, which basically always means that you need a lawful representative. This is usually the king (or another aristocrat), who by extension can do with you what he pleases (which makes dealing with foreigners always an affair of state, so to speak). So if you are a foreigner, the will of the king, for you, is law. Similarly, the king can use foreigners as agents operating outside of the local tribal law.

Within a local society, there does seem to be a 'class' division in society. The most important dividing characteristic in Viking society seems to be the division between the 'free man' and everyone else (the non-free, women, slaves, children etc.). The characteristic of a free man is basically that he can carry weapons. In exceptional circumstances, a woman can also be legally regarded as a 'free man', in which case she would symbolically carry a man's weapons. This is the case when a rich family has no male heirs, for example, but her legal position changes again when she marries. So you can see how legal status is also very much tied with personal status.

There seems to also be a distinction between nobility and commoners, but this distinction is more diffuse, and terminology for 'noble' is different in different periods and regions. This distinction can be seen for example in the amount of compensation that a man has to pay to the family of his murder victims, which is higher for a person with a certain social rank. So while the concept of higher- and lower-status people was relevant for law, it was not universal or particularly solid.

Furthermore, it seems that different areas of law were divided between different areas of responsibility. So where you would be subject to military obligations under a local warlord, you could pay your yearly tax to a different person, and for crimes you would refer to a regional council. A local chief of the army would in theory still be subject to this council, for example. This is also why these 'free men', who are part of the regional council, are so attached to their weapons: these are also the guys who are enforcing the laws. It seems that these law-meetings were held regularly (yearly in most cases), and were in Scandinavia known as a 't(h)ing'. Many places in Scandinavia (and places of Viking settlement) still have names with the word 'ting' in them, suggesting that these were the 'law-places'. At these places, criminal cases and land transactions were done. The idea is that all free men have a say in these, but from the well-known history of the Icelandic althing we know they often did involve factionalism and did get dominated by a few powerful individuals sometimes.