"Like" can take a number of forms, each of which can be achieved in a different way.
One of the most reliable ways to get the players to like/respect an NPC is trust. Set up multiple scenarios where the NPC takes a big risk in trusting the players and having it payoff, and vice versa. This is the classic, "I got your back" situation, used very frequently in movies to establish a partner relationship between two characters.
To make them like/care for an NPC, construct scenes that demonstrate the NPC's vulnerability and provide the players with an opportunity to "do the right thing". The NPC may be injured or asleep, or suffering mental stress and be at the players' mercy. Or, more subtly, the NPC may open up about their emotions, anxieties, or future plans, that they don't want to reveal to just anyone. Or perhaps they need advice from the players as to how to handle some delicate situation.
To make them like/enjoy the company of an NPC, give the NPC some strong (likeable) personality traits, good one-liners, memorable strengths or weaknesses, or so on. The key is to make the players think, "Wow, this guy is... interesting... let's watch him to see what happens next."
All of the above are commonly used in fiction, particularly in multi-hero or action movies. In some cases (often seen in spy movies, cyberpunk stories, etc.) this is then inverted by having the trusted partner betray the primary hero; your players will likely be on the lookout for hints the trusted NPC is going to backstab them, perhaps testing them to be sure. This is also a good way to set up a long-standing arch-enemy (see False Friend, Evil Former Friend, and Face Heel Turn on tvtropes.)
If you want the character to be three dimensional, then avoid stereotypes and look for ways to twist the usual tropes. For instance, understand the "Damsel in Distress" trope and how pervasive it is in modern entertainment, then brainstorm ways of having your female characters act in non-stereotyped ways. Skip the puppies, shoe shopping, and girlish weakness, and avoid killing, maiming, or depowering her; or set your plot up to appear to go that direction, but have it veer off and do something much more unusual and interesting. Don't passively "let the players take her home" but instead have her propose some adventure to the players that gets her some place that betters herself; perhaps she's the only remaining heir to the throne and she's not being considered since she's "just" a woman, but if she had a powerful army under her command, perhaps she'll be the land's first Queen.