I would say the question, as written, is a bit open-ended, as there are many aspects to consider any time you arrange music. I'll try to broadly highlight what some of those issues are, without going in to much depth, to give you a starting point for further investigation.
When arranging music, it's important to understand what the components of a piece of music are. Music will typically consist of a melody line, a bass line, supporting harmonies, and some type of rhythm. Additional components can be added or removed, (for example, adding a counter melody, or dropping out the bass) but this is sort of a standard model. It's also important to have an idea of what your available forces are (i.e. the number of different performers and instruments available), and which of those musical components they can play (good luck getting a bass line out of your ukulele!).
A ukulele is easily capable of playing a melodic line, or it is capable of performing a simple harmonic/rhythmic part. Going beyond these (or perhaps combining them all in a single arrangement) is going to be more difficult, and dependent on the skill level of the performer -- for example, possibly using fingerpicking and arppegiation. I'm not sure what you're specifically after, but a pretty typical simple arrangement would have a singer to sing the melody, then use the ukulele to just fill the harmonies and rhythms (leaving out the bass line altogether, of necessity). Note that in this case, the ukulele does not need to explicitly play the melody, since another instrument (or in this case, voice) is playing it. That frees up the ukulele to just play harmonies/rhythms.
If this is the style of arrangement that you are after, than there are really only two broad things you need to know: what chords to play, and what rhythms to play them with. Choosing a rhythm determines which strumming pattern you use, and can be a matter of personal taste or an attempt to match the original song's style. The rhythm is important to have, to give song its groove, but even a simple rhythm will go a long way.
As for the question of which chords to use, this gets into the question of harmonization. I have a feeling that this is what your actual question may be (how can I select chords to harmonize a melody), but that's another long answer, based on music theory, and isn't specific to ukulele, so I won't go into detail here. If you're playing an existing song, you can often find the chords listed online somewhere. More generally, you'll want to figure out what key you're playing in, and what chords are likely to go in that key. As mentioned in the comments of another answer, not every melody note needs a chord. The rate at which chords change is called the harmonic rhythm, and in a lot of pieces, it tends to be just one chord per bar, with other notes being non-harmonic tones.
If you don't have a voice or other instrument to play the melody, another option is to play just the melody on ukulele. In this case, the question of arranging the melody becomes largely a question of ornamenting the melody line. You can add various embellishments such as rhythmic variations, hammer-ons, pull-offs, harmonics, arppegios, glissandi, and (because ukes don't sustain a tone so long) repeated notes. This gets into performance techniques, and will probably overlap a lot with guitar technique.
A great way to learn how to arrange is to watch other arrangements, and deconstruct what they're doing. Here's an example arrangement I found on youtube of two ukuleles, one playing a simple harmonic/rhythmic style, and the other, an ornamented melodic style. The two performers switch back and forth between styles every once and a while. Note how little movement there is in the left hand of the person who is simply strumming the chords, compared to the person playing the melody line.
Once you've mastered each of these kinds of arrangement techniques, you could go beyond this into more complex arrangements, where you are playing melody, and harmony, and elaborate rhythms. But this will take considerably more skill to arrange and perform. One thing that you will need to be very comfortable with is moving chords up and down the neck, so that you can play any chord in any position along the neck. By way of example, here's Jake Shimabukuro playing a rather involved arrangement. Watch how much his left hand is moving up and down the neck, and pay attention to the way he incorporates various techniques, to highlight rhythms, harmonies, and the melody.
I hope this gives you a good starting point, and happy strumming!