I personal don't grow melons. We don't have the heat units required. and not enough room in the greenhouse. Hopefully someone with more experience will chime in.
Pot size, I'm thinking a 5 gallon nursery pot should work. 1" of water a week, maybe more when fruit is developing.
Most varieties get big and need alot of water. There are a few that are small enough to be recommended for contianer growing, but not many. For those I might try a 5 gal container but to be safe, assuming yours iss a full size variety, go with something like 10-18 gallons. The rubbermaid totes are 18 gallons, and the inexpensive plastic rope handled plastic wash tub looking things at Home depot are the same or larger. The larger the container the less you have to worry about watering.
you can plant them in a container but they will quickly run and spill foliage over the sides and spread...5 gallons would be plenty for the root system.
Best to plant in the ground if you can. Just make a hill of well drained soil and transplant.. give em a shot of H2O when you transplant and some mild fertilizer. let em have at it. You can train the vines if you want to keep them in a certain area or direction.
What kind of melons are they? Like TZ said you would need a BIG pot for almost all melons. I don't think 5 gallons would cut it. The tote size might do, they do needs lots of water. That is until a few weeks before harvesting.
They will more than likely spread ALL OVER the place as well so be prepared for that.
Yes, wherever you put the melons, they will need full sun and fertile soil in order to thrive.
I had read that melons do great in composted manure. So, this year, I put a "reservoir" of composted manure under each "hill" where I am growing melons. I'm looking forward to seeing how they do.