Should one read Tehillim on Rosh Hashanah?
Yes, this is a good practice.
Reading Tehillim is always a positive practice, in petitioning Hashem with the holy words of David Ha-Melech.
On Rosh Hashanah, when we need Divine favor more than any day, it is certainly a virtuous time for Tehillim.
In fact, the Brisk Rav used to say Tehillim during the entire day (outside of prayer time), because he was so concerned for not wasting any time on this great day.