Since we only do N.K. on Yom Tov, and you certainly dreamt a dream since the last Yom Tov, the custom developed that the entire k’hal says this tefilla during N.K. (MB 1)
You should not recite the Ribono shel Olam tefilla on Shabbos unless you had a bad dream on Friday night. (MB 4)
The nusach should be “bain shecholamti al atzmi” first and then “bain shecholamti al achayrim”. (MB 4)
The achronim write that “v’sishm’raini” should be the last word of the tefilla the first time – “us’chonaini” the second time – and “v’sirtzaini” the third time. The Rashal adds that it is good to fit in the third time “v’savvo olai birkas cohanecha” before “v’sirtzaini”. (MB 4)
The Gr”a would say the Ribono shel Olam after the third possuk also. This is instead of the Yehi Ratzon prayer that is printed in the siddurim. (MB 5)
Some have a minhag to always recite the Adir Bamorom prayer when the chazzan begins “v’tov b’ainecha…” in Sim Shalom. (MB 6)
You shouldn’t say the Ribono shel Olam everyday unless you dreamt the night before, because how can you say “chalom cholamti” when you didn’t dream at all? (MB 9)
You can recite this prayer when the chazzan is saying Sim Shalom. Coincide the end with the end of Sim Shalom so the k’hal can answer omayn to your tefilla as well. If you will need more time, you can begin the Ribono shel Olam earlier when the chazzan is saying “Y’vorech’cha” of the N.K. replacement prayer. (Rama – MB 10)