* J'ai les yeux noisette et les cheveux bruns. (This is a very common and straightforward way to say it.)
* Mes yeux sont noisette et mes cheveux sont bruns. (This translates to "My eyes are hazel and my hair is brown." It emphasizes *your* eyes and hair.)
* J'ai des yeux noisette et des cheveux bruns. (This is also correct, but less common than "les". It translates to "I have *some* hazel eyes and *some* brown hair". In this context with eyes and hair it doesn't mean that literally, but just is a slightly less specific way of describing your eyes and hair.)
Important Notes:
* "Noisette" is the most common way to describe hazel eyes in French. It literally means "hazelnut."
* "Brun" is the standard adjective for brown hair.
* When talking about body parts that you possess, French often uses the definite article (le, la, les) rather than the possessive adjective (mon, ma, mes). That's why "J'ai *les* yeux..." is more common than "J'ai *mes* yeux..."
So, J'ai les yeux noisette et les cheveux bruns is probably your best bet.