Become familiar with HBO's body of work. HBO is unlike network television or even basic cable. In order to demonstrate that your show has the unique voice that HBO is looking for, you need to understand what shows have succeeded and been well-received in the past as well as currently. If you can't envision your show side-by-side with other ones in HBO's programming schedule, it's best to start looking for another outlet to pitch your script to.
Try writing a "spec" script for an established HBO show. Cater your script to the tone and style of the show. A spec script is better than a resume in the world of television writing. This not only demonstrates your understanding of the television series, it will help you become familiar with the television script writing format.
Format your script in accordance with proper guidelines. There are many books and resources written on the subject. While the basics are simple to follow they take practice to become comfortable with and longer still to use to the script's advantage.
Get copies of shooting scripts from HBO shows. Get as many different titles as you can and examine them thoroughly to understand HBO's approach to script writing and formatting and strive to apply the knowledge to your own script. Scripts are available from specialty book stores and from a variety of sources online.
Make well-written dialogue a priority. More than special effects or formulaic character development, your script should showcase your ability to write dialogue. When developing your script, be sure the dialogue and interaction between the characters properly addresses the conflict between the characters or their environment.
Do not feel like you have to restrain yourself. HBO is known for being cutting edge and at times a bit gritty. This does not mean you have to go out of your way to be lewd or commercial, but certainly do not shy away from "edgier" subject matter because you feel it may be too taboo. Give your original, well-written and properly formatted story a genuine voice and it will appeal to HBO viewers.