Although the ukelele is often used as an instrument to provide chordal accompaniment (strummed or picked) to other instruments, or commonly voice, there is absolutely no reason why you can't explore other ways to play it, which are not primarily as an accompanying instrument.
For instance, you can play single note lines, as you would on any other instrument, there is an interesting discussion about the right-hand techniques used for this on this forum page.
Of course, if you want to play recognisable pieces/songs solo on the ukelele, it will probably be more interesting if you can combine a chordal and melodic approach. The simplest way to do this, would be to learn a variety of chord shapes for each key and chord type, with different chord tones at the top. This would allow you to play the chords for a piece/song, with the melody notes at the top (of course, a little bit of extra study would be needed to find fingerings for non-chord tones in the melody above the chords).
In order to get started with learning more than just a few ukelele chord shapes, and so open up greater solo possibilities, it would be a good idea to find out about ukelele notation and technique. A good resource for this (and I just have to mention it, due to it's slightly naughty name!) is the amusingly named UkeHunt website. This has some great info about ukelele technique and notation. Also, there are a lot of TABs for pieces in a wide variety of styles on this site. Many of these illustrate the approach I outline above; they have a melodic line accompanied by some amount of harmony. Other pieces are far more complex, resembling the kind of pieces played on classical guitar, for instance. And then there are also short, relatively simple, single line riffs and melodies.
By all means learn to play chords on the ukelele, whether you choose to sing with it or not (and you may wish to accompany other instrumentalists and/or singers anyway), but there are plenty of resources available with a little bit of looking online, for learning to play single line melodies and pieces with more complex textures, too (melody and accompaniment or even contrapuntal).