What is the best way to get an Exotic or Legendary Engram in Destiny Crucible?

The easiest way to get Exotic engrams from Crucible (and from PvE) is to pop a Three of Coins. This will give you a buff that gives you a vastly increased chance of getting an Exotic the next time you finish a Crucible match or kill an "Ultra" in PvE. However, the buff is used up whenever this occurs, even if you don't get an Exotic. Thus, you'll need to use one every round in Crucible.

Every time the buff is used up without giving you an Exotic, the chance of getting one next time you have the buff is increased as well, up to a certain cap that Bungie hasn't given us the formula for.

For Legendary engrams from the Crucible, you'll just have to wait on the RNG to bless you. Do note that they were pulled from the Iron Banner rewards list this time around:

Removed Legendary Engrams from the possible end-of-activity rewards list, creating a greater chance to receive Iron Banner specific items

Iron Banner specific items include Legendary-rank gear, but you won't find Engrams you have to decrypt.

Probably the easiest way to get an Exotic is to do the arena in The Reef every week and you WILL get a Legendary from that.

The best way to get exotics or legendary engrams is by playing Challenge of Elders because of the boss drops and the chest you get when you win. The next is playing raids, there are an expanse of hidden chests and bosses that will drop goodies. Second try using crucible boosts, which will increase the chances of exotics or legendary engrams. It works because I got an exotic engram. If that does not work try winning matches not losing in pvp because you get better rewards if you win. Also try to level up your ranks for the crucible so you can get prices from shaxx in the tower.