This is a territorial battle, and the incumbant cat is winning, so you need to give your cat a helping hand.
Find where the neigbourhood cat sprays or defacates, and scrub the area thoroughly with hot water and bleach, rinse and repeat. This will stop him automatically re-marking.
Place some mothballs all around your garden boundary, especially on fences and walls, but leave an area around your back door.
This should discourage the other cat from marking, and allow your cat to mark outside. once he has done so a few times, he should settle, and use his tray.
If you see the other cat, chase it off as quick as you can, and don't allow any urine of faeces to remain any longer than you have to.
It will take time, but once your cat has reclaimed his own area, he'll be quit content to defend it - it's because the other cat was there first that he's having problems.