stick a volt meter across the terminals when the engine is off , should be 12.5 volts or thereabouts, if so the battery has charge. I think you should do this while the engine is being started as well and get another reading to see what the volts drops to under full load.
start the engine and use the volt meter again, it should now be 13.5 or so, if so the alternator is working and the battery is charging.
AA vans have the kit in them to do akll this for you as well.
That'll be £35 + VAT please oh and by the way your exaust needs replacing and you could do with 4 new tyres, disks and 4 sets of brake pads, would you like me to fix those things while you're here?
Paul