Hippos, you're being given a hard time, and I wouldn't be surprised if you'd stopped reading, but in case you haven't...
I really understand the urge to kick your DS out. You did actually say upstream (though most people seem to have missed it) that you don't really want to throw him out, you just want a break. There are some of us here who have lived with difficult, unlikeable teens, and know how badly parents sometimes need a break. If you don't get one when you need one, you can get quite desperate, can't you? I guess that's the point you're at...
Your family has been through some difficult times. I feel for you all. Your son has had to deal with a lot of changes, many of them beyond his control: your divorce, not living with his dad any more (and it sounds like perhaps not seeing him either), coming to terms with you having a new partner, leaving boarding school... You say he is self-harming, so he is obviously struggling.
I might be wrong, but I guess you are caught up in a powerful, confusing cocktail of emotions: I can imagine you might be angry, guilty, bereaved, frustrated, upset, exhausted, resentful, worried, and more - all at the same time. If you think that's true, it might be worth asking yourself if really it is these horrible feelings you want to go away, rather than your son?
He is probably also feeling a similar nasty mixture of emotions. It may be making him behave badly. I know my own son behaves badly when he's angry and upset; many people do. It's likely that his emotions and yours are clashing a bit...
In a similar situation with my own son a couple of years ago, I discovered some useful things...
Firstly, like Maryz says, you can't change anyone else, only yourself. If you focus on trying to change your DS's behaviour, you will probably fail, but if you aim to change your responses to his behaviour, you stand a good chance. Secondly - importantly - I discovered that I needed to be kind to myself. No-one else was, except occasionally (that's not meant to sound self-pitying; I think that's true for very many mothers, esp single parents), and it turned out that kindness was a sort of rechargeable energy, so while I wasn't receiving enough, after a while I wasn't able to give out much... I wasn't unkind, but nor was I able to be very kind, tolerant or compassionate - especially towards my son when he was behaving like an arse. But once I started being kind to myself, and my 'kindness batteries' got sort of recharged, I found myself much better able to be kind to him, and more likely to react with sympathy and understanding rather than frustration...
Also, incidentally and quite amazingly, being nice to myself seemed to encourage other people including my son to be nicer to me, as if I'd suddenly given them the message that I was worth kindness and respect. :)
All of this may be totally off the mark. Just ignore it if it doesn't ring true. :) But maybe it'll help... Good luck. :)