We all like different things in a wave kite depending on ability, riding style, conditions etc.
I think Peterheirman pretty much nailed the usual requirements though, drift, relaunch etc are a given.
Turning speed is debatable, some like fast turning kites that they usually turn a lot and follow on the board, some like slow turning kites, these are usually the guys that like to park and ride in good sideshore conditions.
I am a bit in between on this I like a kite that turns fairly fast after a fair amount of bar input, I am not to fond of the crazy fast turning at the slightest input kites but would still probably prefer that to a real slow turning kite.
I notice a few people say they don't like a kite that pulls through the turn, again I think this is debatable.
My previous kite used to do a very fast with not much power sort of stall/pivot turn then shoot across the window, the kites I am on now have a much smoother pull through the turn and I love it.
Some people when trying the same kite complain that they are being pulled of the board through the turns but this is mainly due to the fact that they are using a kite 2 or 4 metres bigger than required.
If like me you like to ride strapless and use as much of the power of the wave as possible so use the smallest kite you can get away you would probably love a kite that has a constant pull through the turn.
As far as upwind ability goes, for me personally I rate it very highly, yes board design plays a significant part in upwind ability as well but when you combine a board and kite both with good up ability your wave count goes through the roof.
I have sacrificed a slight bit of performance on the turns with my board due to flatter tail rocker and concave vs v but combined with my kite that is extremely efficient upwind I am rarely beaten/outpointed getting back to the peak.
I used to do a lot of downwinders but rarely do these days as it is just so easy to get back upwind to start another run.
I think upwind ability also equates to fairly good speed, wave riding is pretty much all I do but we have an annual Redbull 19km open ocean race that is starting to attract some of our nations best racers that come armed with Ozone Edges, Airush and Cabrinha race kites and custom/race boards.
I haven't got the money or desire to spent thousands of dollars on that sort of gear for one race a year so just use my wave kites and whatever board I can find (this year I used an ancient 7'6" Naish Sky Pirate that cost $65 from a second hand store ) , yet still usually manage to be in the top 10-20% I think due to the efficiency of the kite.