It’s not offensive to be blunt about a character’s race. You’ll want to work it in naturally, though, as you would any telling detail in your story. For example, something introductory along the lines of these would be perfectly fine:
She was African American, the deep brown complexion she inherited from her mother, though her eyes, dark as smoke, she definitely got from her father.
He pegged her as African-American, with the rich coloring of golden copper.
She shifted in her chair, feeling awkward. Being the only Black girl in the room, everyone’s eyes instantly locked on her as the professor… [etc etc]
It really depends on the moment and the story’s style how you’d like to introduce race (especially if race doesn’t exist as we know it in your story). There’s certain aspects to consider, though, such as narrative voice.
Here is a book passage that clearly and naturally introduces the race of the main character:
Most people would chalk me up as your basic offspring of a mixed-raced marriage. I had skin people of poetic bent called Café au lait and which I called, to myself at least, ‘my daddy shagged a White woman’ brown. I had in-between hair, a thick umber mane that resisted dreadlocks, braids, and every other form of styling. And I had Dad’s features, or so my Aunt Aggie always told me, which I liked; all of Dad’s pictures pegged him for a looker of the Denzel Washington variety. -Faerie Blood, Angela Korra’ti
This paragraph is jam-packed with indications of the character’s race (and also, character voice). We learn she’s Black and White not just because she mentions being the offspring of a mixed-race marriage, seeing as that can be a mix of anything, but because she makes the “daddy shagged a White woman” jest, notes how Afro hairstyles don’t work with her, which builds the assumption of her being part Black, and then compares her father to a Black celebrity.
[However: Note the above does use a food description, albeit a bit mockingly, but food-skin comparisons are something generally to avoid.]
Here are some examples of descriptors of the character’s race threaded throughout the story. This is from Fire Baptized by Kenya Wright.
“Heart pounding, I stayed close to the buildings, hoping my brown skin would blend with the night’s shadows. Cold rain dripped into my eyes. My wet dreadlocks fell onto my face, sticking to my cheeks and blocking my view.” (Skin tone noted in comparison to settings, hair noted in similar fashion as the elements affect the character’s hair)
“You’re a sexy Mixie, aren’t you? Dark, like chocolate. You Haitian?” (See! This was a major creep. Noo choco-comparisons no! Anyhow, someone else noting the character’s race, another method)
“I don’t find that funny.” I backed away from him and crossed my arms around my chest. “Especially since i’m Black. Cut to the chase, and tell me what’s going on.” (Situational instance of noting race)
So you see, there’s no one way to introduce a character’s race and there’s definitely nothing wrong with being blunt. Be creative about it, and as you would with any story, and spot out those natural openings where character description would be appropriate.