Specifically, I would like to know what kind of permission a British explorer would have needed to find the Northwest Passage (ex. would they have needed a charter?). I would also like to know how they would have received funding for an expedition and if they were all military-led. I know there were plenty of expeditions done through the royal navy, but I'm not sure if these were the only option and if they were government-sponsored, would the government only use ships from the navy?
A fella called Sir John Ross relied on the funds of a gin magnate called Felix Booth for his second Arctic expedition of 1829. Paid for the ship too! He even got the Navy to pay his men extra after being stuck in the ice for years. And many of the expeditions after the disappearance of Sir John Franklin and his men were at least partially paid for by his wife while the navy covered the rest.
The ships they used varied. Some expeditions used ex warships while others used modified second hand ships or, very rarely new ships built to survive the Arctic.
All I can say about how they would present it; They'd rock up to the admiralty with all their planning and requirements, mention any financial aid they were receiving and hope for the best. Best to be friends with a chap named Barrow!
This was off the top of my head so there may be errors, reckon it's pretty bang on though. I've got a heap of books relating to Arctic exploration and am happy to look stuff up fer you if you want, bit of a hobby of mine!