For most experiments F6 is good enough (like transplants), but to be sure most of us do until F10. You can also increase your odds by doing genomic PCRs that allows you to pick the offspring with most off the BL6 traits.
I'm not sure of the specifics of the coat color determinants in mice, but I agree with Steven. You should plan your back-crosses with respect to residual heterozygosity conferred by simple genetic principles and not phenotypic markers alone.
You can do the math - If an animal inherits 50% of their genome from their parents, then in a back-cross their residual genomic heterozygosity is (1/2)^n after n generations of crossing to another inbred line. The resulting "% new background" is therefore 1-((1/2)^n) for n generations of back-crossing, yielding the numbers presented by Steven.
I usually refer to these generations as BC1F1, BC2F1, BC3F1, etc. since they aren't filial generations per se - every back-cross generates a new F1 in the Mendelian sense.
According to a colleague of mine, some of the larger mouse colonies/venders offer genotyping panels to more quantitatively assess back-crossing efficiency - might be worth looking into.