Checking Intonation

How is the most simple way to check the intonation of my uke? Can I fret the 12th fret and check the tuning to see if the intonation is spot on?

That is the simplest way. Just compare the tuner reading of the open string and the fretted 12th fret. NB: Don't fret too hard or you can bend the note sharp at the 12th.

The other way is to compare the note fretted at the 12th fret with the harmonic produced by touching the string over that same fret. If the fretted note is flat compared to the harmonic then the intonation is flat (i.e. string length between nut and saddle is too long). If the fretted note is sharp compared to the harmonic then the intonation is sharp (i.e. string length between nut and saddle is too short).