NB: This answer is written considering things as of season 1, since the OP hasn't seen past that yet I've included post-season 1 information in a spoiler tag.
She is alleged to have murdered both of her parents so if convicted she would be double murderer for that crime, the typical definition of a "serial killer" is a person who murders 3 or more people over a time period spanning more than 30 days so assuming that many days have passed before she kills Henry Madsen in S01E04 (or failing that by the time she kills DCI Ian Reed in S01E05) then she would fit that definition.
Her conversation with Luther on the bridge strongly implies that she is guilty of her parent's murder but as she points out to Luther he has no way of convicting her since the evidence he has is of no value in an official legal proceeding against her since it's circumstantial and was obtained illegally to boot.
She has committed various other ancillory crimes since the murder but generally speaking they are all part of the steps she takes to conceal her part in the various murders so the headline is that she is a serial murderer, specifically one who kills people for convienience or personal gain, what is termed a "Comfort killer" as opposed to the more commonly depicted tropes of a "Thrill killer" (for whom the act of killing itself is what drives them) or a "Mission killer" who believes they are doing the "right thing" in killing certain types of people (e.g. believing that homosexuality is so wrong as to require the killer to elminate gay people)
Subsequent to Season 1:
Subsequent to her esacpe from a mental institution in Season 2 (which Luther aids her in) she claims to have killed at least one other person - her husband in Berlin but this happens off screen so we have no direct evidence that this took place although she has no real reason to lie to Luther about it. This further supports the notion that she is someone whose sociopathy has lead to her becoming a comfort killer.
PS: I know way too much about serial killers, I should probably get out more!