The basic intonation check has to do with checking the bridge saddle and nut positions, and the effective scale length of the strings.
First play a touch harmonic on a string at the 12th fret - the string will sound an octave higher than its open pitch. Then fret the same string at the 12th fret and pluck the note. The harmonic and the fretted note should be just about the same pitch.
If the fretted note sounds flatter than the harmonic, then the scale length is too long. If the fretted note sounds sharper than the harmonic, then either 1. The scale length is too short, or 2. The action is too high, and fretting the note is pulling the string sharp.
On a fixed bridge instrument like most ukuleles, fine-tuning the scale length means adjusting the saddle and/or nut to lengthen or shorten the distance between them. Lowering the action means lowering the saddle and/or nut (or nut slots). These adjustments should mostly be left to a competent luthier or technician.
The other common intonation problem is a too-high nut, which pulls the intonation sharp on the first few frets. You can check this by fretting each string at the 1st fret, and checking the notes (g#-C#-F-A#) with an electronic tuner. If they're more then 10 cents sharp or so and you can't easily push the pitch down with finger pressure toward the bridge, then the nut is too high. Again, a competent tech can lower the nut.
It's probably uncommon for frets themselves to be in the wrong positions, because they're laid out mathematically. But if so, all the notes at a given fret would be too sharp or flat compared to other frets. Then you've really got a problem that an experienced luthier should look at.
Hope that helps ...
