Run a populist, pro-tenants campaign. You wouldn't have to spend tens of thousands on television or radio ads. Nor break the bank on city-wide mailings. Just pound the pavement and target high-density neighborhoods with a lots of apartments. Rental housing would be harder to track down, but it could be done to a certain degree.
People know what neigborhoods and streets have a high percentage of rental housing.
Write off wealthy residential neighborhoods. If someone ran a campaign targeted primarily at winning the vote of tenants, he or she wouldn't even have to win over a majority of voting tenants to capture a seat. And maybe turnout might go up among tenants because many who might not otherwise vote would be eager to get a non-property owner onto the city council. Maybe tenants would vote in several tenants. It wouldn't have to be a totally one-dimensional campaign, however.