Arena keys and corresponding wins

Solution 1:

Champion key is 8 wins.

Straight from a Hearthstone Wiki

Wins    Key         No.**   Possible rewards
0       Novice      2       A pack and 25-40 gold or dust or a common card.
1       Apprentice  2       A pack and 30-50 gold or dust or a common card.
2       Journeyman  2       A pack, and either 40-50 gold or dust, or a common or rare card.
3       Copper      3       A pack and 25-35 gold, plus either about 20-25 dust or gold, or a common or rare card.
4       Silver      3       A pack and 40-60 gold, plus either about 20-25 dust or gold, or a common or rare card.
5       Gold        3       A pack and 50-60 gold, plus either about 45-60 dust or gold, or a common or rare card.
6       Platinum    3-4     A pack and 75-85 gold, plus one or two prizes from the following*: about 50 dust or gold, and regular or golden common or rare card.
7       Diamond     3-4     A pack and about 155 gold, and one or two of the following*: about 25 dust or gold, and common, rare and golden cards.
8       Champion    3-4     A pack and about 155 gold, and one or two of the following*: 25 dust, 20-40 gold, and a regular or golden variation of any card.
9       Ruby        3-4     A pack, about 155 gold, and one or two of the following*: 50 dust, 75-95 gold, and a regular or golden variation of any card.
10      Frostborn   3-4     A pack, about 180 gold, and one or two of the following*: 70-90 dust, 65-115 gold, a golden common, and regular or golden variation of any card, rare or above.
11      Molten      3-4     A pack, about 200 gold, and one or two of the following*: 60-90 dust, 80-180 gold, a golden common, and regular or golden variation of any card, rare or above.
12      Lightforge  5       A pack, about 230 gold, and three of the following*: another pack, 20-25 dust, 20-175 gold, a golden common, and regular or golden variation of any card, rare or above.

* Each reward can appear multiple times
** Number of rewards