How does trustworthiness with NPCs work

the-elder-scrolls-v-skyrim

In light of Raven Dreamer's answer to Why can I sometimes take half of a shops items while the oher half is considered stealing?, how does one get people to like the player-character? In other words, get their disposition higher (towards the player-character) and get them to allow the player-character to take their stuff?
Edit: While rereading kotekzot's answer, I wondered how (easy it is) to affect a relationship negatively.
Edit 2: Then I realized that I was focusing solely on take vs. steal when I could be asking about what else that trustworthiness (as I'll call it instead of relationship) affects. If it is felt that this edit broadens too much; that what trustworthiness affects deserves its own question, this question can be changed back.

Best Answer

The only legit way to increase an NPC's disposition toward you is to complete quests for them (or somebody close to them, in a few cases). Not all quests will increase disposition and not all NPCs will have a quest for you.

You can also use setrelationshiprank player 1 in console to befriended the targeted NPC. Read more about console commands.