Some people don't like dealing with relationships and feelings beyond friendship in their table top RP. That's fine, though this post may not be for them. Others, like myself, rather enjoy the depth a more significant relationship can bring to a character's story. This relationship doesn't have to be romantic, but it can be. Either way, there are some common things you want to make sure of before you just try to dive in.
As Always, OOC Communication is Key
Before you jump into an IC relationship with someone else's PC, you want to make sure that you and the other player are on the same page. Sometimes characters take actions that one player (or their character) reads as meaning one thing, while the other meant it in a totally different light. Talking to the other player lets you iron those things out. It also lets you make a clear divide between you and them as players, and your PCs as characters.
The better the players know each other outside of the game the less important this step is, but it is still good to have. Beyond just making sure both (or however many) parties are involved are ok with a relationship happening, you also may want to establish things you aren't interested in doing, or things you're hoping to get out of it. For example, Jeremy may be looking to deal with the consequences when his character's dalliances with people outside the relationship causes drama and tension, but Michael may not be interested in a story where his character gets cheated on and he has to deal with that emotional fall out in his fantasy life. Alternatively, if both Sarah and Jennifer are really digging finding how a relationship helps their characters find stability, but also pushes them to more and more extremes as they try to defend their S.O. then that's great.
Either way, real worl relationships throw people into enough loops. Talk about the fantasy ones a bit before going in for it.
Some people don't have the same table comfort as others. I personally am not good at handling deep or strong character emotions if I'm not allowed to summarize actions or things that the character would say. Others may not know how one acts in a relationship while they are also regularly going into life or death scenarios. If someone is uncomfortable, give them the time and space - or whatever they need - to be comfortable. Make sure they're still cool with whatever the plan is. Make sure the other people at the table are cool with what may be going on.
Keep It IC
Sometimes the two PCs getting into a relationship IC are also in a relationship OOC. Sometimes they're not. Either way, my advice is to keep the relationships separate. It's great if you and your girlfriend/boyfriend want to RP adventurers who are dating. However, if you blur the lines between the IC romance and the OOC romance you're going to have a bad time. I've seen and read too many horror stories where people in a relationship have had real problems, or even broken up or divorced, because the divide between their IC love life and OOC love life blurred too much and a fight that started in the game broke out into the real world.
On the other hand, if you aren't in a relationship with the other person OOC you need to remember that it is IC and IC only. While often in younger players (at least that people admit to) I've also seen lots of people become flirty or smitten in real life because of a game, and sometimes it works out well but often it doesn't. Mostly it comes down to expectations, but it can also come down to just social norms. Adam may be happy to RP someone interested in Clara's character in game, but that doesn't mean Adam wants Clara to be following him around after game, flirting with him, or pushing into his personal space. Jeff and Annie may be hitting it off in character, and Annie might think Jeff is a cool guy and friend out of the game but that doesn't mean she's looking for Jeff to be a romantic partner, and even if she is, it doesn't mean the OOC relationship is going to follow the same rules as the IC one.
Essentially, keep the IC dating IC. Fake emotions for fake people are one thing. Real emotions for real people are something totally different.
Have Fun With It
Don't think of the relationship as work. Have fun with it. Indulge in it. See where it takes the character. Relationships can be one of the easier parts of a character to understand because even if we don't know what it's like to professionally hunt 50' tall fire breathing lizard cats for a living, we do know what it's like to form an emotional bond with someone. Let that run. Let it fuel some of your actions. Talka bout it with the other end of the relationship with it. Roll with it. See what happens. Find out what really matters to your character.