There are many factors that come in to play regarding getting a course built. PDGA and Disc Golf Association both have some great resources on getting courses built on their websites. It helps to get in contact with local parks and recreation offices to plant the seed. PDGA has a sample course proposal you can edit to meet your needs. Submit that along with some ideas you have. This would include locations, costs, benefits of the sport. Go in to the meeting (if you get one) with a professional attitude about the sport. Encourage the benefits and let them see your passion for the sport.
Be firm but flexible about your ideas. If your goal is to put in an 18 hole championship course and the only funding they have is for a 9 hole course then weigh your options for the course. Decide what you are willing to compromise with. Things such as concrete tee pads and multiple pins are great, but can come later. Consider where your funding is coming from. (grants, tax dollars, donations) Oftentimes you can get in-kind donations, volunteer labor and other things to help get the course built.
The city may not see money back for the course in a pay-to-play aspect(unless that is your goal), but may see an increase in out of town folks coming to play the course and spending $ when they are in town. The city may also see a benefit of investing a sport that gets people outdoors. Emphasize that an 18 hole course can accommodate more than 100 people at one time. This is much more than a baseball field, soccer field, or playground and can cost much much less.
Create a long term goal for disc golf and the city. If DG is something new to the area then you may have to start small to gather interest for the city to invest more $ in to the sport. Work with local clubs (if any), schools, and businesses to generate interest and $.
My experience/ time line went like this: 1. Presented a proposal to the city's parks & rec advisory board three times with increasing details once initial plans were approved. 2. Presented the proposal to city council to gain approval. 3. Get the neighborhood approval. 4. Presented proposal to donors and grant foundations. 4. Design and build the course. 5. Installation of the course. 6. Improvements.
We were forced to concede to a 9-hole course to start with. The course has increased in popularity in the year since it has been open. We are in the early stages of presenting the next stage of our DG proposal to the city which will include an 18 hole championship course. We expect that to be built in the next 5 years.