Laws of "Devarim Shebekedusha" - Part 2 (Kaddish d'Rabbanan)
It is the accepted custom to recite Kaddish (called “Kaddish d’Rabbanan”) after learning Torah — whether after a session of study of the Written Torah or of the Oral Torah. So too, this Kaddish is said after the public reciting of Tehillim. Ideally, this rule applies only when the above events are done with at least ten men present, thus constituting a tzibur, a congregation.
The Taz requires there to be ten people present only when saying the Kaddish after learning or reciting verses, even if there were less than ten people for the actual learning. However the Magen Avraham requires there to be ten people present for the learning as well and this also seems to be the opinion of the Gra. Therefore, in the event that Torah learning or verses are read without ten people present, Kaddish should not be said afterwards when there are ten people, unless a mizmor or some verses are first recited (Mishneh Berurah 55:2). The Piskei Teshuvot further explains that if Tehillim was recited, verses should be said, and if Torah was learned than the Mishna of Rabbi Elazar or Rabbi Chananya Ben Akashya should be said.
The Mishneh Berurah writes: “Just as it is good to decrease the amount of blessings one recites (i.e. that one should not make “extra” blessings when they are not really necessary), so too one should decrease the amount of Kaddeshim said. It seems that the intent of this instruction is not to say numerous Kaddeshim after reading verses of Tanach, Mishna or Gemarah. However, saying one Kaddish at the completion of learning Torah, or after reading chapters of Tehillim, is a great and wonderful thing, and this is also an obligation according to the law. (Piskei Tshuvot)
The Birkei Yosaf, authored by the Chida writes in the name of the sefer Vayakel Moshe that there is nothing better to protect against the threat of mazikin (spiritual damage) like reading Tehillim, and thus children who are within the year of mourning for the passing of a parent should say Tehillim, while reciting a Kaddish afterwards daily. (Brought in Kaf HaChaim 55:26)
There is also a well-known Midrash of Rabbi Akiva which explains the great benefit to the soul of a deceased person when Kaddish is said by his children. According to the story recounted there, through the power of Kaddish the soul was able to rise from gehinom and enter Gan Eden.
