Begin by reviewing some of Gwen Stefani's older work with No Doubt. Note how her vocals were extremely stylized in her early career, and how she used this style as a basis for her current singing style.
Practice basic singing. Work with a vocal coach or through self instruction to learn proper technique, as this technique must be learned before it can be obscured. Try to singing scales and on your range as Gwen Stefani has a fairly high vocal range.
Sing along with one of Gwen Stefani's songs; note that most of her most vocally challenging work was done on No Doubt's first couple of albums. Try to sing a ballad or slow song to really get a feel for Stefani's vocal style.
Pay special attention to Gwen's vibrato style and her playful way of phrasing each verse. Master her child-like vocal quality and try to sing in a strong yet youthful manner, occasionally dipping into a whisper or an angsty type whine.
Start singing along with different Gwen Stefani songs, and master all the vocal nuances of that particular song. Record yourself singing like Gwen Stefani and compare this with her singing.