I am unaware that there is any prescribed order to this. Most places I have attended do the following order:
1 - Announce the names of those who are ill and that Tehillim will be recited for them. I am assuming that you are talking about a small (1-3 people, avg.) group of people, though, there is no strict protocol. Also, I have usually seen this announcement for "newly" ill people, someone undergoing a serious procedure, or someone who's situation has changed to a dangerous near-death situation. (I.e. - someone who was already ill for some time, Tehillim was already said, and is now getting better and recovering at home, might not be announced.) Also, if you have 50 names, I don't think they would announce all the names prior to Tehillim.
2 - Tehillim is recited responsively
3 - Prayer for the ill (Mi Shebrach L'Cholim is recited with the Hebrew names mentioned aloud. I have never seen anywhere where the names are silently said.
The procedure is different, however, if you are talking about the Mi Sheberach L'Cholim prayer if it is recited as part of the Torah reading service. There are various customs in each shul.
Some have people line up before the Gabbai, and each person tells the Gabbai the ill name(s) which are recited aloud. My shul used to do it this way for about 60 years until about 8 years ago, the rabbi realize that this caused a crowd at the Bimah and some people were embarassed to announce their ill people's names aloud so that others could hear. As a result, the procedure has been (and this is done in many other shuls) to have the Gabbia recite the beginning of the prayer, and pause, and each person would individually recite the names, silently. (As a Gabbai, myself, I have a master list of names which I recite silently during the pause. Yes, there might be some duplication between my recital and some congregants, but that's not a problem.)