TL;DR about Joker and how to RP Him (Or write him in Fanfic)
If you want to RP Joker, and you don't know Alan Moore's The Killing Joke (Know the story, not necessarily having read it; I know not everyone can pick it up), I would seriously reconsider apping the character. Basically, even if that's not the background story you use (Say you're writing cartoon or movie Joker), the insights into the character apply across the board, for any version of the character.
The most important aspect to role playing Joker is to pick a persona for him. Joker doesn't have one personality through the stretch of his canons. If anything, he has connecting traits-- some sort of performance aspect to everything-- but even the types of crimes that he commits aren't consistent. One decade, he might do steady bizarre but violent pranks. In another story arc, he steals a nuclear weapon and becomes part of the UN. In yet another arc, he puts broken glass under his finger nails and uses it to cut someone's face up.
Pick a persona. Does he see himself as an artist? A ringleader? A stand up comedian? A sight-gag extraordinaire? Or, if you're going for a more modern Joker, is he mainly a fucked up sadist that paints his face and giggles? Pick one of those and stick to it, at least until you feel comfortable with the character. It is canon for him to switch personas (check out some of Grant Morrison's work, where he actually argues that Joker has no main personality, and actually constantly reinvents himself; Morrison wrote a comic explaining this and describing the moment of personality change), but he doesn't do it every couple of days. He does it slowly, after many struggles with the superheroes.
When you've picked the persona, think about how violent/sadistic that persona is. He's always sadistic, but his ability to do actually damage face to face is debatable. Some writers write him as physically weak, and that's why he can't actually fight using anything but his elaborate weapons and gags (one punch from pretty much anyone can fell him), and others write him as a very capable fighter. If he's capable of a lot of personal damage, he tends to be very very hands on, barging over to henchmen and killing them, or walking past henchmen to go after Batman and others himself. If he's not...well, then he's going to talk to Batman in a very different way, isn't he?
Also, I would note that Joker is VERY difficult to play as a player character for the sheer fact that it's difficult to play Joker when he can't lash out and poison/blow up/shoot/torture a crap load of people]
Unlike the persona, which stays the same for a long stretch of time, his moods are entirely erratic, short lived and, most importantly, EXAGGERATED. If he's pouting, it's a huge, face-stretching pout. If he's laughing, it's the funniest shit ever. If he's angry, he's shoving someone to the floor and kicking them in the face and stomach, and then maybe shooting them for good measure. If he likes a person, he'll take them under his wing for about two seconds until they piss him off and he goes apeshit on them. If you are playing him without an exaggerated mood (and his default is actually AMUSED, not charming-evil as some people play him), you're probably doing something wrong.
Joker can be played two very separate ways. One way, which is plot wise the easiest, but may be much harder to maintain, is that Joker NEVER has a plan. He thinks of everything on the fly. He may be as aware of his next step if his plans fail as the reader or the cops are. The second way is that he's ALWAYS thinking like, 19 steps ahead (Batman thinks 20 steps ahead). This involves a lot of planning and player communication, and really means that random interactions with Joker are rarer.
Also...if you're playing with a Harley...Joker doesn't actually love her. Sure, there are some exceptions, but he spends a LOT of time giving her 'missions' in hopes that she'll die on the way. He romances her only when it suits him-- to escape Arkham, to keep her from killing him-- but not out of a sense of love. He pulls her in, and then throws her away, constantly. Keeping her either close or dead is his major choice.
In the same vein...sure, you can make the argument that Joker has homoerotic undertones in regards to his obsession with Batman, but really? Even if they exist, they aren't on the surface, and they aren't likely something that he registers.