First, look at what is available in your price range. I would say that a boat in your price range would most likely need around $20,000 or more to fix it up. More if you have no experience and will be learning as you go.
Can it be done? Look at some of the contributors to this forum and you will see it can be done. How much? No idea. Find a boat first and then you will have a better idea of what it would cost to fix it up.
Are you a US citizen? If so, what skills do you bring to the table for working outside the US? If you expect to work illegally, you are either competing against the non-skilled labor of the country you are visiting, or depending on selling your services to other yachties. Keep in mind that the US is kinder than many/most countries. The US doesn't put people in jail when they're caught working illegally. Other countries aren't so considerate, particularly those countries where they don't feel an obligation to feed their jailed "guests".
Have you considered signing on to crew on other boats to get you around the world? Helps build your skill set and is a lot cheaper. Easier to bail out if it's not for you or your friend.
Lots of questions, not many answers for you this early in your dream. If you haven't done it already, take the time to read more about cruising and voyaging on this board. It might help.