I doubt that you've got any realistic chance of claiming against the seller (especially if the car was sold without a warranty).
I'd also question whether it's actually an alternator fault. I've had the alternator pack up on several cars and it's always happened while I've actually been driving. The chances of an alternator providing enough power to get you home but not enough power to leave the battery charged would seem to be fairly slim to me. It's far more likely to be a duff battery, which won't hold sufficient charge overnight.
A new battery would probably only cost around £60 to £80, with free fitting, if you phone around for quotes. If you actually need an alternator, a breaker's yard should only charge around £30 to £40 for one. Search here:
http://www.breakeryard.com/
My local garage would probably only charge 1 hour's labour (£48 inc VAT), at the very most, for fitting it.