For monks chanting the traditional hours… they would chant all the psalms in the course of the week. (That’s why the five decade Rosary has 150 beads – it was a sort of Psalter Lite.)
However, I presume that you’re not planning to chant five or seven times a day, including midnight. So if I were you, I would sing all the psalms in the course of a month. You would only have to chant 3 or 4 a day, and that would be plenty to think about.
Look under “Daily Office” or “Liturgy of the Hours” sites. They will have a lot more prayers and readings than just psalms, but they’ll give you a good start on the Psalms.
divineoffice.org has a good listing of stuff for the day, as does universalis. There are also some Latin sites.
Vatican Radio broadcasts Lauds, Vespers, and Compline every day, from the Vatican or from alternating monasteries and convents. This will help you with tunes.
www.divinumofficium.com is a great resource. As you may know, since ancient times the Church has promulgated some kind of prayer book based on reading the psalms. For most of its history the psalter was reading throughout a one-week period, with special prayers according to the season. I hope it’s just what you’re looking for.
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