There is no “right way” for reading through the Psalms other than asking the Holy Spirit to guide you and speak to you through them. However, for the past several years, I have read through the book of Psalms during the 50 days between Resurrection Sunday/First Fruits and Shavout/Pentecost as part of my devotional reading plan. This fifty days is also known as the Counting of the Omer during the Feast of Weeks. As there are 150 Psalms, I read three each day, beginning with Psalm 1, 51, and 101 and ending on the final day on Psalm 50, 100, and 150.
You don’t have to wait until specific time to begin. Read through the Psalms any time you want to set aside a time to draw close to God. Each morning as you begin, thank God for his goodness and faithfulness and for his gift of Jesus Christ, claim the atonement of the blood of Jesus and ask him to renew your mind, invite in the present of the Holy Spirit and ask God what he wants to reveal or speak to you about that day . . . and then begin.