Figure out what key your piece is in, and analyze each measure to see what chords are in it. In jazz pieces, the chords will already be written out for you. Runs at their most basic are built on scales, and what scale you use for a run depends on the chord that is playing below it. If you see an F major chord, you will probably want to base your run on an F major scale.
Start out improvising runs by moving slowly from chord to chord, singing up and down on the notes of that chord's corresponding scale. Sing ascending notes on one measure and descending notes on the next.
Add flavor to your run by adding rhythm. Instead of singing all the notes in a run at the same length, try swinging the notes by dividing them into groups of two and making the first note in each group slightly longer. Or you can alternate groups of five or six very fast notes with single, longer notes.
Change the notes. Once you are comfortable with using scales to create your runs, you can branch out and use different notes or different scales entirely. This takes practice, and in the end it comes down to what sounds good with the style of music you are playing. Many musicians use blue notes in their music. This technique involves bending some of the notes so they are slightly flat, or lower than usual. Most of the time, blue notes are on the third or fifth degree of the major scale.
Split long runs into manageable sections and practice each section by itself. One logical place to make these divisions is where you would normally breathe. You can also practice each measure separately.
Practice your run with staccato, or short and choppy notes. This will help you memorize the pitches, and when you eventually sing the run smoothly, you will be much more accurate.
Analyze your run to find where the stresses are, then practice your run with emphasis on those stresses. This will help you get a more musical approach.
Work to expand your vocal range by singing your highest note, then going above it. It won't sound good at first, but over time, you'll find that your highest note will get higher. This will help you to sing runs that cover a large range.