You may want to contact the landlord first. Once established, its very difficult to remove a grape vine, they root like crazy. Landlord may not want a vine growing up their deck that they can't get rid of with anything short of a truck with a tow chain.

Also, odds are you would be getting the vine fairly small, and they only grow at the rate of a few feet a year. By the time you could get it to grow up high enough to reach the full sun of the deck and start producing fruit, it may already be time for you to move again. Grapes can take upwards of a decade to really mature and get in their stride and can live for well over a century, they're not really a good choice for a fast growing temporary plant.

Perhaps you could try something a little faster growing and a little less permanent? Maybe a blueberry bush in a large container? Those will last years, give you lots of fresh fruit, and best of all when you move you can take them with you.

What about just growing one in a large container?

Usually not recommended, again because of the massive root systems mature grapes tend to grow. From what I've seen, a minimum of 6'/1m of soil is required for healthy grapes, with the roots easily able to grow twice that deep or more while spreading out like a fan.

Basically, a grape vine big enough to climb up to that deck is going to have a root system like a small to medium sized tree.

Fruitnut grows them in containers....maybe he'll share his results with you or you could search the forum

If you want a vine but find out you can't have grapes, since they anchor to walls, if I recall. You might look into passion fruit vines, they grow fast, produce fruit, and offer coverage.