Do you have the variance application?
You can save yourself the initial trip to City Hall if you get the form (available online?), fill it out, and bring it with all requisite attendant information. This form and additional info include:
Your name, the address of the property in question, the current zoning, your zoning change request, the owners names and addresses for all adjacent properties, and possibly photos.
If you can't get the form online, your first trip to City Hall will be to meet with someone from Zoning department who will give you the forms and explain what you should do with it.
Once you fill out and submit the paperwork, they should send you a notice of when your case will be heard (4-6 weeks after you submit your application). Usually there are monthly or every-other-monthly meetings dedicated to variances. They will notify everyone who could be affected, and there will be a public notice stating "4700 Elmira Rd, bounded by Banks and Olympia Street, requests change from 2-family housing to 4-family housing". Anyone who cares will show up at the meeting to support/contest your variance.
The city will also have their own opinions about it, and will discuss the situation with you at the meeting. Bring any supporting documentation you can: Examples of chicken-friendly zoning verbage, a list of major metropolitan areas and their basic chicken codes, newspaper articles about urban chickens, letters from all your neighbors saying they approve (if they can't make the meeting in person), your personal coop plans (including location in your yard, and location of neighboring houses), and your plan for keeping things clean and rodent free.
I suggest sitting in on a meeting prior to your scheduled meeting so you know what to expect. The discussions will be very different, but the questions will be similar, and you'll get a feel for what information they'll be looking for.
Good luck.