I'd first check the easy things like is the boot light or any other interior light staying on then check that the main battery connections are tight and in good condition also check the earth strap to the engine / gearbox.
Next I'd check the current drain aspect, the best way to find what circuit is causing the excessive currant drain is to fit a current sensing multimeter in series with the positive battery lead on the car and the battery, or use a current clamp meter on the battery positive cable.
A typical current drain with everything off and un-alarmed should be around 0.05Amp and if its higher than this remove a fuse and re-measure, if measurement doesn't change then replace the fuse and remove the next.
Once you've isolated the circuit with the excessive drain check the wiring connections and plugs to components and the wiring harness that runs by the engine bay fuse box - the wiring harness chaffs / rubs through and gets damaged - but its not always easy to see so be careful when checking..
Also check that the alternator is actually charging the battery in the car by taking a battery voltage reading with the engine off then another with the engine running. The 2nd reading should be higher than the 1st and climbing steadily. And also check that the alternator is secure to the engine as this provides the ground for the charging circuit and make sure that the engine / gearbox earth strap is secure and the contacts are clean and bright.
if the alternator appears not to be working - take it to a electrical specialist and ask them to test it as sometimes you can replace just the diode pack and this will repair the alternator without requiring replacement