I myself am a keyboard player, though I often play guitar. I have no trouble playing one note trill with three fingers, it's much harder with 4 fingers but all 5 is so hard. How to train to be able to do it? Any good exercise? How should the hand move in this case? I mention both left hand and right hand practicing.
"Doctor, it hurts when I do this!" "Well, don't do it then!"
Your thumb is stronger than an individual finger, and has a completely independent set of muscles and tendons to move it. The muscles and tendons of your fingers are all interconnected to some degree - that is most obvious on 4 and 5. So use your thumb in preference to "only the other four fingers", unless it is playing another note.
You don't have to use adjacent fingers. You might find 13131313 works better than 12121212. (Move your thumb sideways, so it stays under the 3). Also, your thumb doesn't have to play the notes "on the beat". 31313131 is not the same as 13131313!
For a long tremolo on one note, fingerings lie 12131213 or 31213121 are less tiring than repeating two fingers. Without the thumb, try patterns like 23432343 or 23242324.
Since you said "keyboard player" not "piano player", you might want to try a technique that was used on early keyboard instruments like the harpsichord, where the "playing point" of the note is close to the start of the key movement, unlike a piano. That was to "lock" your hand in a fixed position and shake it from your wrist, using only one finger to play the notes. That doesn't work well on an acoustic piano, because you have to press and release the key too far to play each note. It might work on synth keyboard, though. On a well regulated harpsichord keyboard, the notes will sound reliably when the keys are pressed only 1/4 of the way down.
Fast trills are easier to play on grand pianos, as their mechanics lend themselves better to key recovery. The keys actually don't have to be let fully up before they can strike the strings again.Keyboards vary considerably, so you may find yourself trying on an unsuitable one. That said, trills are mostly two not affairs, unless you mean tremolo, where the same note is played quite quickly by two or more digits.
A useful idea is to practise by using fingers 4 and 5, then 3,4 and 5, until you incorporate the strongest with thumb, and 2.
After considerable thought, I find that what you are trying to achieve is somewhat pointless. It has no real practical applications, as tremolo can be achieved better using the two or three digits normally featuring in this activity. Some techniques can't be transferred from one instrument to another with any degree of success - or practical uses.
My advice would be to practice those fingerings slow. Since this technique is used to play repeated notes very fast one is usually tempted truing to play them fast when the fingers are not used to this kind of movement.
This example is taken off Carl Czernys Practical Exercises for Beginners (Op.599 Nr.12)
Number 1: this snippet is definitely the easiest and has nothing more to do with your question then demonstrating where to start from. (As reference: You will find a fingering like this in the C.F. Peters edition, revised by Adolf Ruthardt)
Number 2: here you got a fingering for practicing the 5-4-3-2 change on one note. (As reference: You will find a fingering like this in the G. Schirmer edition, revised by Giuseppe Buonamici)
Number 3: I would recommend something like this to start with for using all five fingers on one note. The change from 1-4 is done the same way as if you practice for instance a descending C major scale over at least two octaves.