I agree with cloisters.
What you are doing initially is considering setting up a Private Association of The Faithful. This is covered in Canon Law and we are quite free to set up private associations of the faithful. Initially, as cloisters pointed out, governance needs to be worked out - also formation once a person joins is very important and in this I am thinking of formation in The Faith (Catholic Catechism for example) and Spiritual Formation and on an ongoing basis. Not absolutely necessary, but I would be searching for a priest chaplain for my private association. His function in one's association also needs to go into one's Rule and statutes if one decides to have an actual chaplain. It does not HAVE to be a priest - but in my personal experience in my own diocese, religious can be pretty off the planet and into all sorts of questionable matters. Not so much our priests, although it can happen.
Heaps of other things too that need to be worked out between the foundation members - lots of groundwork.
Were it me, once I feel I have something to present to my diocesan leadership (Rule of Life however one goes about it), I would send a letter to His Grace with the rule enclosed as that of a private association of the faithful advising His Grace that what is enclosed is a courtesy to his authority as my diocesan leadership and authority - and to advise him of what is taking place in his diocese and why or a short covering letter to the Rule. A little history of what has taken place.
If you Google "catholic - private association of the faithful" you will find heaps of reading matter about forming one. should rather fully inform you on this move.
Private Association of The Faithful
What I would also do is once the Rule and statutes etc. etc. are all agreed upon by your group, contact diocesan offices in your dioceses and ask to whom you need to speak (it can differ from diocese to diocese) re a private association of the faithful and what is the proper procedure to advise His Grace as a matter of courtesy of what is taking place. What sort of documentation would be best. If you get a curt reply stating you don't HAVE to advise His Grace - state that you wish to do so in order to keep him informed re his diocese. Then I would just send direct to His Grace what I thought would be appropriate.
And I would not do this until the group had settled on their Rule etc. and had been living the way of life for at least 6 months. It is only a courtesy to His Grace, not a requirement.
Expect plenty of teething problems. Heaps of them! This way if they come along you will not be surprised. If they don't come along, you will also be then most pleasantly surprised remembering to give thanks and praise to The Lord where it belongs.
Expect problems as you go. Our Bishops are under that much stress always in one way or another. Something new and untried can give then a near on anxiety attack. Not always of course, but expect it. Sometimes a Bishop can take it all in its stride, but then opposition can come from all sorts of quarters, even from within your association.
"Take up your cross and come follow me" What an honour to be thus called and we all are but Grace and The Lord are with us every single step of the way in His Indicative and in His Permissive Will. If you don't understand this teaching of The Church, familiarize yourself with it - in these posts into Catholic Discussion sites can provide excellent info and leads, they can also lead one astray, as can Google. Ensure Google info is from sound and reliable sources and this is very important to check out usually in the About facility on any website. And if I can't find anything reliable and soundly Catholic about a site, then I leave it alone.
That is all 'off the top of my head' as I type. Other members might post and be far more helpful than I.:thumbsup: