In order to get the color, another breed must be introduced. Most use black to cross to self blue and all the resulting offspring would be black split to lav. Those f1 birds could either be bred together to create 25% visual lav, 50% split lav, and 25% black, but this would create a problem in which it would be impossible to tell the blacks from the split lav as they would both be visually black, so f1 could be bred back to the original lav bird to create 50% lav, 50% split lav offspring. The original lav bird could be retired as the color is now set in the crosses. From there, the f2 lavs can be bred to black silkies to yield all split lavs. The f3 could be bred back to the f2 lavs to get 50% lavs and those can be bred back to a black silkie. This would go on until a desired result is reached. It takes several years to get a new variety established in a breed.
If you wanted to take the easier approach, you could always order some of the self blue silkie chicks Ideal offers and just start breeding them to nice blacks to improve the type.
Heavy culling would be necessary in both cases.