Listen to a recording of the music notes. If this is part of a song, search online for it. Even a small sample of the song is helpful in trying to figure out the harmonic progression.
Play the notes on the piano. Get a good idea of the direction of the melody. If there isn't a key written for the notes (rare, but possible) see if you can figure out what the key is. A helpful way to determine the key is to play through the notes and, at the end, sing to yourself what the resting final note would be. Our brains are trained to unconsciously search for finality in music and resolved phrases. By singing the music to yourself, this allows your brain to meander through the music and land on the correct notes.
Play the notes in the right hand on the piano and try to follow along with chords in the left hand. The chords should be basic, three- or four-note chords--nothing too complicated. It's a bit of a guessing game at this point, although there are a few music theory elements to keep in mind--mostly that the first chord is going to be the "I" chord (that is, a chord based off the first note in the key's scale). Generally, though not always, the second-to-last measure in a song will be a "V" chord. There tends to only be one chord per measure.
Tinker around with the chords in the left hand and notes in the right hand. When you feel confident in a chord write it down on the paper. Continue to check your chords to see if they sound right. When writing out the chords, the general harmonic tab notation is writing the first letter of the chord name in capital letters. If it's in minor write a lower case "m" next to it (for example, C-minor would be written "Cm"). Dominant seventh chords, like "V7" chords, are written as the chord letter with a 7 after it (G-dominant-seventh is written "G7"). Major seventh chords are written with a "maj7" (C-major-seventh, "Cmaj7").
Check the harmonic tabs that you figured out against any online tab databases (see the Resources section). Be aware that many of these can be incorrect and you should almost always trust your work against the work of an anonymous other. But it's a good way to see if your work has matched up with others.