תהלים is something that is an important part of our lives year-round, of course. However, at certain times there might be additional attention to it.
But how should it be approached? Does הקב"ה need more lip service, עבודת הפה rather than עבודה שבלב?
The answer is emphatically no. Rather, כוונת הלב is what is sought and needed.
Rav Avigdor Miller z"l, in last week's תורת אביגדור booklet (p.7), gives some guidelines for its proper reciting. Say it slowly, with understanding and emotion (click to see what Rav Miller brings from the חובות הלבבות, and more, under the headline "How to Say Tehillim", on p.7 there).
Elsewhere, in the excellent biography of Rav Miller, it is related that in his youth in Baltimore he would go to a Shul (although not explicitly stated there, based on other sections of the book, where it speaks about Rav Miller's youth in Baltimore, I think that the simple assumption and understanding is that it was a Litvishe Shul - if it was otherwise that would likely have been noted IMHO) where the men said Tehillim together with a beautiful melody (p.43). On Shabbos after סעודה שלישית they would sing the entire chapter 119 (תמניא אפי, the longest kapitel in Tehillim), the saying of which was a widespread minhag in ליטא, and considered a סגולה for חשק ואהבת התורה.
May we be זוכה to meaningful recitations, and כתיבה וחתימה טובה.