In ordinary singing, the singer generally keeps the tongue flat and the fundamental is the only clearly audible pitch. To do the whistle-like style of overtone singing, the sides of the tongue are curved upward and held nearly against the upper premolar teeth - creating a seal with the roof of the mouth all the way around (with a small opening for air to pass). With this in mind, try singing "errrr." For higher overtones, move the tongue forward.
Vowel sounds and lip shapes are important in fine-tuning the overtones. The lowest overtones are emphasized with tight "oo" sounds, while increasingly higher harmonic overtones can be heard as vowels change through "oh...awe...ah...ay...ee," and everything in between. Another approach to overtone singing is using these vowel shapes without the modified tongue position. This is especially effective for the lower harmonic overtones of the series (2-7) and one can resonate harmonics 3-5 strongly by nasalizing the vowel transition from awe to ah. Good mnemonics for remembering the sequence of vowel shape transitions are the words "why" for emphasizing overtones ascending the harmonic overtone series and "yow" (descending).
We encourage you to listen to overtone singers and throat singers to try imitating what you hear.
As you progress with overtone singing, you will find that more refined control down near the base of the tongue and in the throat allows one to amplify and isolate each desired harmonic overtone more effectively than just emphasizing tongue or vowel techniques (so the tip of the tongue needn't move to change to the desired overtone, for example). Abdominal breath control becomes critical as one develops more demanding styles of Tuvan and Mongolian throat singing.
Try overtone singing each harmonic through the series with your mouth completely closed and keeping the fundamental as steady as possible. Imagine the vowels shapes of "why" (ascending the harmonic overtone series sequentially) and "yow" (descending) while doing this. Harmonics 2, 3 and 4 are often difficult to hear in this way, but it's excellent for developing subtle listening skills.