Qatar is an Islamic country and Qataris dress traditionally. Expats and travelers are expected to wear modest clothes and to avoid mini skirts and tank tops. There is no need for women to cover their hair. And you won't get arrested or fined for what you are wearing, but you may get some stern looks from locals if you are wearing beachwear in a shopping mall.
Bikinis are allowed in hotel pools and beaches, but generally not recommended for public beaches. There is no specific dress code for expats or travelers, but it is expected and recommended to be sensible especially in traditional or public places.
Have fun, enjoy the sun and don't fret over it.
Can tourists wear traditional Qatari clothes? Here is the answer from Qatar's most well-known YouTube personality addressing that question in this clip.