Most singers will not be able to perform the chromatic scale or any of the scale variations without first gaining some margin of ear training. One way to train the ear, and the voice, to the chromatic scale is to play every key on the piano and sing along. Start with an octave your voice is comfortable with and work up or down from there. Stop when you can no longer comfortably sing a note or when the you cannot match a tone. Software is another ear-training option. Solfege is free ear training software for PC and Mac (see resources) that allows singers to utilize the entire chromatic scale in a variety of exercises designed to train the ear and the voice to recognize tones and semi-tones.
Use the entire chromatic scale to perform expansive vocal runs. This is a difficult challenge for all but the most accomplished singers, but practicing such runs can develop the voice quickly and effectively. Expansive vocal runs using the chromatic scale require singers to begin on one note of the chromatic scale and sing every note in ascending and descending order, including all sharps and flats as well as the octave, seeking to achieve a smooth transition between each note.
Singing arpeggios in common chord progressions is an effective method that helps develop control of the voice as well as the ability to hear tonal variations and musical progressions. An arpeggio is a broken chord; a chord progression is movement between chords in specific keys. Singers often practice singing chords built from the chromatic scale based on common chord progressions. There are numerous chord progressions possible, so you may want to purchase a basic music theory book to keep on hand while you practice. You can find one at a music or book store, or online at no charge. In addition to common chord progressions, you can practice singing chords built on every, or every other, note in the chromatic scale in consecutive order. This is difficult to master, but can effectively develop extreme vocal control.