George R. R. Martin had already created some words in the language, but it was David J. Peterson who, based on the existing words, created the rest of the language with a team of people. As of May 2015, the language has a vocabulary of over 4000 words, Peterson wants to increase it to 10,000 in the future.

When interviewed about its creation Peterson said that two major factors played a role. One of which was the topography of the area. It would help answer questions about what their lives are like, who they interact with and how those interactions are shaping their culture.
Technology is the second factor. The world is set at a much lower level of technology than our world, but the Dothraki are even below that level. They have words for what they need in their lifestyle, but may borrow words from others.

Not having sailed over the ocean, and not having the desire to do so, meant that they never created words like ‘boat. Instead the Dothraki refer to boats as ‘wooden horses’.

Peterson produced a book called Living Language Dothraki for people wanting to learn the language. There is also an unofficial fansite where people have made an online dictionary https://docs.dothraki.org/Dothraki.pdf 

Peterson got his inspiration from languages as Turkish, Estonian, Russian, Inuktitut and Swahili.
When creating the language, he had to keep two things in mind: use the already existing words in the book and make it easily pronounceable and learnable for the actors. Due to the use of dental consonants, Peterson says the language might sounds like a mix between Arabic and Spanish.The Dothraki do not have a writing system, but if they were to get one it would be based on the writing system used in the Free Cities.
The language is mostly inflectional. There is also a clear word order. Just like in English it’s subject-verb-object.And just like in English, when there is only a subject, the subject will precede the verb.

https://en.wikipedia.org/wiki/Fusional_language