When describing a character I usually provide some indication of their race/ethnicity since a lot of my work has multiracial characters and I tend to use elements of ethnicity in characterization. Most of the time I find a tasteful way to describe the color of their skin and start them in a scenario that allows them to show their heritage compared to other characters eg. name, how they speak, where they're from, what they wear and like etc.
I've recently had someone read a particular scene involving a character, and was surprised afterwards, when they told me that they didn't understand why the character was speaking a certain way (slang etc) and had completely missed what their ethnicity was and that this was why they were speaking this way.
My point is that you can never predict the level of cultural awareness of a reader, and so certain aspects of a characterization will fail to convey the character's ethnicity if that reader has little to no knowledge of that particular culture, which will affect that reader's enjoyment.
When comparing books and film for example, I find it interesting that many film scripts I have read are very up front about character ethnicity eg. Trent, 21, Black. But this is rare in books and I'm not sure why.
So with all that said what are some ways to convey a character's race/ethnicity clearly (while hopefully steering clear of stereotypes) but appropriately (without sounding racist or offending anyone)?