Probably more that one way to do this, but this is my go-to method anytime I need to memorize something:
1. Read out whatever you want to memorize in full (most Psalms are short). You want to memorize as much as you can handle in one go, so try not to chunk unless it's one of those huge Psalms (119

). When you need to chunk, try to make the chunks as big as you can handle. E.g. For Psalms 119, I'd do verses 1-24 or 1-32 for my days work. Read your target a couple times until you feel you know what's coming after each line.
2. When you feel ready, cover up the Psalm & speak it out loud or clearly in your head. If you spot mistakes or struggle, go back to Step 1 & check your trouble spots. When you feel you've got it, go to Step 3.
3. When you feel like you can recite the whole thing, write out your target from memory. This'll provide a record of what you got right & the few spots you struggle. Testing yourself in this way is the most important step. Do this writing on a paper, no typing or phone.
4. Check the incorrect spots & write them in where you made a mistake. Don't re-write the whole thing during error correction because that takes too much time, just cross stuff out & fill in blanks.
5. Repeat the process until you can write out a Psalm perfectly. When you go back to reading your target, you'll naturally pay extra attention to the parts you had trouble with. Writing stuff out by hand is pretty tedious, so you'll be really determined to get it right within 2-3 tries & not start the Step 3 writing test before you can recite it & hold it in your mind.
5. Last step, when you can write it out perfectly, cover up your writing & recite the Psalm out loud one more time from memory, Uncover the verses that you wrote as you go so that you can compare on the fly. This step is because when you're writing out the Psalms in Step 3, you'll be writing at a slow cadence, but you'll want to be able to speak the Psalms quickly like second nature. This will also get rid of small errors like replacing "the" with "a" or similar.
6. In the future, you may have some minor touch ups (e.g. using the word "which" when the word is "that"), but they should be very easy to correct & you shouldn't need to write the whole Psalm out again.
Small tip: If you listen to audio recordings of Psalms, try to listen in the same version that you want to memorize in. Not a huge deal, but this just helps you not to 'combine' the two versions by accident. I tend to goof on this & recite a verse which is half-KJV & half-NIV
In 2013 I memorized 2000+ Korean words from a dictionary using basically the above with an added step of Korean --> English, & English --> back to Korean. I'd memorize 30-60 per day & the next day I'd consistently only forget 1-3 words (5%) & this was easy to correct because of the previous day's focus on nailing the writing step perfectly. Scripture, prose, or instructions are significantly easier because there's a theme or direction to the work.
The core process of Read until I feel I can recite --> Recite until I feel ready --> Writing to test is what works for me. I don't claim this method is the best & it certainly takes work. However, once Step 5 is done, the results for me have lasted years, excepting the tiny & infrequent corrections mentioned in Step 6. This would absolutely be my go-to technique if I needed to memorize Psalms, although I don't want to knock anyone else's method.