As a first step take a look at this British Embassy document -
Applying for a UK Passport Whilst in the Philippines
It's an 'oldish' document but the documentary requirements will be a big help
Then review these newer links:-
Overseas British passport applications - Philippines
Applying for a passport from outside the UK - guide to the online application service
Applying for a passport from outside the UK - Supporting Documents
Apply for a British Passport - Welcome to the online passport application service for British nationals
This online system is still very new. I haven't personally gone through the online process so can provide any hints/tips etc but as far as I know the online application form can be saved and returned to for changes before final submission.
Hope this helps