The viking as depicted in TV (or literature in general) is mostly a literature phenonemon and should be understood with a grain of salt.

We don´t really fully understand how slavery worked inside Scandinavia, where the economy was mostly based on agriculture and hunting. We know it existed because there are several legal texts on slavery that have survived. But archeologial evidence has not revealed a lot of what we might have expected. (We find iron shackles in Viking areas in Ireland, but not in Scandinavia.) Yet we know that a slave would have shared quarters with there masters because the houses at the time were just one big hall without seperate rooms.

The Icelandic sagas or not really usable as historical sources for the viking age. But since you really seem to be asking about a TV viking, rather than a real one, they are sort of acceptable as sources to answer this question, and most of them feature a strong personal relationship between a slave and his owners as a plot device. In Egils saga Skallagrimssonar, a fallout between Egill and his father results in each of them killing each others favorit slave. In Brennu-njals saga, Njal promises his slave Thordur that if killed he will only settle for a free mans payment. In Laxdæla saga, Olafur Pa, the son of a female slave and her owner becomes a major chieftain - but it also turns out that his mother was the captured daughter of an Irish king. (Olafur is mentioned in a lot of sagas). There is also plenty of instances where a slave owner gives land to his slaves in the settlement period of Iceland. So it´s not necessarly out of line with viking literature for a slave to rise through the ranks.

There may however be problems with the idea that vikings raided to gather slaves. From the slave traders perspective it´s a bad business model to raid yourself because the space is limited on the boat and not every stranger you pick up in haste is a suitable slave. A better business model is to buy preselected slaves where the less usable or troubleseme individuals have been weeded out, since the profit comes from good end price, rather than low cost. Slavery was not unique to vikings at the time, and they actually were more of businessmen than raiders.