I am finding the basics a bit of a struggle too at the moment. Things that have helped me are:
If you don't feel able to have a shower, at least stand at sink and wash face and under arms. Get some dry shampoo (batiste) and use that if your hair looks awful. After a while you start to realise a shower would take just as long, but this seems less daunting, and you don't need as many towels. You still need to shower every few days but this helps you feel more presentable in between.
Do a little bit of something if you feel the whole task is too daunting. So don't think "Oh god I have ALL this washing up to do" just think "Okay, I'm going to reduce this washing up pile." and do one or two sinkfuls of the stuff nearest the sink, or think "Right we will need 3 plates, 2 saucepans, 6 forks, etc for today" and just wash those up, then if the water is still clear do whatever else is to hand.
Are you eating? I don't remember to eat when I'm feeling down so I have got loads of ready meals and other easy food like frozen veg, waffles, fish fingers etc in the freezer so I don't have to think about it. They tend to be quite bland too so I can just eat it without thinking about it. The other thing that helps is to buy food I really really like and actually want to eat. DS gets fish fingers/chicken nuggets and frozen veg most nights but that's ok, because at least I know he will eat it.
Get a big calendar on the wall and write reminders to yourself like which day the bins go out, recycling, etc.
Type yourself a big list of these 5 daily essential tasks, put some clip art of a big cup of tea or slice of cake or whatever you want as a "reward" at the end and print it out and put it in a plastic sleeve thing,(or laminate if you know someone with a laminator) stick this up next to the calendar. You can then tick off/cross through when you have done all the jobs, using a whiteboard pen. You can do similar for children if they are old enough.
You are doing fab and it's a really good thing that you are posting for support
Now I need to get off my arse and follow my own advice!