You get the Trials of Osris entry invite by getting consistent wins in the crucible. There's not a specific amount, jusy keep on winning. Once you have the invite you use strange coins as the entry fee. 3 losses and you're out, but all wins count towards Osris reputation which can net you guns, shaders etc etc...
I really want to do this, but it's a pain without a decent team as some of the randoms are that bad it makes me genuinely suprised they can get the disc into the ps4 on target. Anyone else up for it?