How do I spectate a game I am in?

Solution 1:

According to this,

Considering it’s an online multiplayer game, it’s surprising to see at its launch lacks a dedicated spectator mode. But Blizzard does make it possible to spectate in Overwatch, although it isn’t as convenient as a dedicated spectator mode.
One way to spectate a game is to attempt to join a game a friend is currently playing. If the game is full, you’ll be forced to sit on the sidelines, but as a result, you can spectate your friend as well as all of the other players in the match. If your friend continues with the same group into another match, and there’s no vacancies, you’ll continue to sit in the new game as a spectator.
Another way to spectate a game is by joining a new game yourself which also has all team spots filled. When this happens, you’ll be forced to enter spectator mode, although as soon as a spot frees up, you’ll be thrown into the action. Unfortunately, there’s no way to back out of the game to re-enter spectator mode.

and this is inline with what I have experienced through in-game testing.

Solution 2:

Currently there is no official way to spectate a game, but it is being implemented soon. According to Jeff Kaplan here.

Although if you die while in game, I believe you can switch between watching the kill cam and other players on your team, which would be the closest thing to spectating, although you'd only have it for 6 seconds.