Roxy likes to poke her nose in other peoples' business, even if it ends up doing more wrong than right: for example, forcing Jane to confess her feelings to Jake, and blowing up Jane's computer.
She's also more than just an annoying drunk who likes to interfere with people. She's lonely. She probably doesn't interact much with the carapacians who live near her - half of them are hostile, after all. She has a load of cats, possibly to fill the void of not having any real-life friends. The reason for her alcoholism probably stems from her loneliness. It mirrors Gamzee's sopor slime addiction; he started doing the stuff because his lusus was never home. Roxy's mother was never home because she died years ago, so she probably started drinking to feel less lonely.

Her peppy personality is probably just a facade, like how Dave's coolkid facade isn't his 'true' personality. We've seen bits of Dave's true personality, for example that one conversation with Terezi when he finds Bro's dead body, so we may see this happen to Roxy soon, perhaps if she sobers up at some point in the story. Like Dave also, she is very clingy, but unlike Dave, she shows it freely - she checks in on Jane after Jane failed to tell Jake how she felt and ruined her chances of being with him, and was disappointed and slightly angry when Jane told her this. This is all happening while everything around Roxy is on fire - she was presumably fighting the robotic drones. The fact that she went to the trouble of checking to see how her friend was doing during a fight that she could possibly end up dead in shows that she is attached to her friends and would literally die for them.