To be honest, I am not sure this is going to end well. People rarely change their ways - whatever the reason, health, personal problems, sheer laziness, it's unlikely you'll be able to fix this just by talking to her. You may be able to achieve some smaller things e.g. her doing the washing up and throwing her food away but a complete personality change overnight is unlikely. Is she OK emotionally? Has she always been this way or has something changed recently? I hate to offer internet diagnosis but moodiness, lack of self care and seeming inability/neglect of small personal tasks like the washing up are classic signs of depression so she may not be able to help the way she is?
What is your living situation, ultimately do you choose who lives in your house or would it need to be the landlord as I can see one of you moving out being the only eventual solution?
When you say 'we' how many of you is this - if more than a couple of you I would nominate just 2 people to have the conversation, preferably the ones she gets on with best. Any more than that and it'll be overwhelming and seem as though you are ganging up on her.
As to what to say, I guess it depends how kind you want to be, you could either have an 'Are you OK, we are concerned about you, maybe you should seek help' kind of conversation or a more factual 'In order for us to live together we need you to abide by XYZ' type of one. Either way I would start by saying something positive if you can about her as a housemate, that you enjoy living with her or similar, before launching into your complaints/concerns. Try and keep things calm and factual and give a few concrete examples, e.g. the washing up, rather than things which can be construed as more personal, e.g. the lying, or a massive overwhelming list where it seems like you object to everything about her. I'd pick 2 or 3 things max which are your biggest concerns and tackle those. Let other things slide for now. Maybe suggest some 'house rules' re things like the washing-up and a rota for cleaning which everyone agrees are reasonable, e.g. dirty washing up to be left no more than 24 hours (I know some people would find that unacceptable and would want it done within a few hours but it's a compromise you have to make living with housemates IMO).
I'd be aware that she almost certainly will be upset, and may hit back with counter accusations or accuse you of bullying or whatever, this is a normal response when you challenge someone's way of living, particularly if they have other things going on, I just don't think there is a particularly nice way to have that conversation...