The Shulhan Aruch (Orah Haim, 119:1) says that if one wants to make a personal request in any of the middle blessings of the Amida, one may do so on behalf of oneself or one’s family. For example, if one wanted to recite a personal prayer for one’s livelihood, one could insert it in the blessing of Birkat Hashanim, as that is the blessing which discusses one’s livelihood. The same is true if one wishes to pray for the health of oneself or a member of one’s household, one would insert a personal prayer in Refa’enu. If one wishes to pray on behalf of someone outside of one’s household, one could insert the prayer in Shome’a Tefila or after the second “Yihyu Leratzon” at the end of the Amida. In this context, Rabbi Ben Zion Abba Shaul (Or Lezion, vol. II, ch. 7, § 33), based on the Gemara (Shabbat 105) says that a Torah scholar is considered a member of one’s household and one would be able to pray for his health in Refa’enu itself. He also adds that one should not get into the habit of inserting personal prayers every time one prays lest they become rote and thus less meaningful.
The HIDA (Birke Yosef), based on the Arizal, says that the proper course of action is to think about the health or livelihood in Refa’enu or Birkat Hashanim, respectively, and then to recite a verbal prayer for those things in Shome’a Tefila.
Another detail is that when praying for someone’s health, one should say the person’s name and their mother’s name (eg. David ben Mazal, Tamar bat Simha). As well, when praying for one’s father one should add the honorific “Avi Mori” (lit. “my father, my teacher”).
Summary: When praying on behalf of oneself or a member of one’s household, one may insert a personal prayer in the appropriate blessing in the Amida. For other people one should do so in Shome’a Tefila or after the second “Yihyu Leratzon”.