There is a custom to recite ten chapters of Tehillim each day during the month of Elul after the morning Shaharit service. By reciting ten chapters each day throughout the thirty days of Elul, one completes the entire 150-chapter book of Tehillim twice, reading a total of 300 chapters. Three hundred is the numerical value of the word "Kaper" ("atonement"), and is thus a very significant number as we approach the judgment on Rosh Hashanah. The custom is to continue reciting Tehillim even after Elul, and complete the book a third time during the Aseret Yemeh Teshuba – the Ten Days of Repentance from Rosh Hashanah through Yom Kippur.
It should be noted that while this is certainly a laudable practice that we should all endeavor to observe, one must ensure to recite the text slowly and with precision. Often, in a person’s rush to complete the customary quota of ten daily chapters, he reads the text incorrectly, mispronouncing words and skipping letters or syllables. There is an important principle that when it comes to voluntary prayers such as this, "a little with concentration is preferable to a lot without concentration." Quality is far more important than quantity. Therefore, if a person is unable to spend the time reciting ten chapters of Tehillim each day with precise, proper pronunciation and concentration, it is preferable for him to recite fewer chapters correctly and with concentration.
Summary: It is an admirable custom to recite ten chapters of Tehillim each day throughout the month of Elul, and to then recite the entire book of Tehillim over the course of the Ten Days of Repentance. One must ensure, however, to recite the words properly and with concentration, and it is preferable to recite fewer chapters properly than to recite ten chapters improperly.