What order do I get work done in?

I would get the boiler moved into the correct position first, then have a general builder come and rip out the door, knock bricks out and install the patio, then have the kitchen/floor fitted. If any old kitchen units are in the way of the new boiler position or the wall you need the patio space, you will just have to remove a few and live with less cupboards for a few weeks

I Think your first port of call would be to have a structural engineer design your new opening. If you are Knocking through a supportive wall for a new set of patio doors then assuming that the wall is 2 story you would need a steel in place. A good builder would then be able to help you achieve this including the rip out of the old leading to a free space for the plumber to move the boiler and neaten the pipe work, tiler in next if you are tiling the entire floor then carpenter for the kitchen finally finishing with a decorator. Rather than organising everything yourself a good main contractor will price for everything and bring his trades in for the job. I hope this was helpful.