Is it normal to get a rash on your chest and legs if you have pink eye?

No. Normal pink eye can come with cold like symptoms but shouldn't cause a rash. See your pcp or a dermatologist to see what is causing your rash.
Not typical. Perhaps the pink eye is an allergic reaction, not a virus/bacteria, and the rash is part of the allergy. Rarely we could be dealing with a devastating disease called steven johnsons. If you were very ill with a systemic infection, perhaps you could see this.