The best way to read the psalms is also the most common way: to make these ancient prayers your own and speak them directly to God. So many of the poems catch deep human feelings that you can't help being moved by them.
But not all the psalms seem attractive. Some sound harsh, self-congratulatory, or boring. You will not find it easy to pray these until you understand them. And there are so many psalms! This is the longest book in the Bible. To compensate, many people read only selected psalms, skimming over the others. But then they miss the deep messages found there, including the messages that the New Testament writers saw when they quoted Psalms more than any other Old Testament book. The richest lessons from Psalms may come from particularly difficult poems you must read again and again until you begin to see what the author had in mind.
The original Hebrew of these poems probably used no rhyme or strict rhythm as traditional English poems do. Instead, the psalmists wrote with parallelism, following tone thought by a "rhyming" thought or by its opposite. Fortunately, this kind of poetry can be translated into almost any language without loss.
Readers may be confused by the psalms' frequent change of voice. In a single poem the psalmist may talk to God, then talk about him, and then return to talking to him, all in rapid succession. This would be strange English prose, but was common in Hebrew poetry.
Because so many of the psalm titles refer to David, you may find it helpful to refer to his life story. It is found in I Samuel 16-31, the whole book of II Samuel, and the first two chapters of I Kings. However, most of the psalms can make perfect sense without reference to any outside information. They merely ask - and reward - time and close attention. Read and reread them. They grow richer with careful study.

An Overview of Psalms in 7 Days
Day 1. Read the Introduction to Psalms, and Psalm 19: Perfect Harmony
Day 2. Read Psalm 23: The Shepherd Song
Day 3. Read Psalm 27: Confidence in Trouble
Day 4. Read Psalm 51: The Great Confession
Day 5. Read Psalm 84: The Delight of Worshiping God
Day 6. Read Psalm 103: God's Benefits
Day 7. Read Psalm 139: God's Care