Answer by taking a quote from the following article:

Kaine was born at Saint Joseph's Hospital in St. Paul, Minnesota. He is the eldest of three sons born to Mary Kathleen (nee Burns), a home economics teacher, and Albert Alexander Kaine, Jr., a welder and the owner of a small iron-working shop. He was raised Catholic. One of Kaine's great-grandparents was Scottish and the other seven were Irish.

After graduating from law school, Kaine was a law clerk for Judge R. Lanier Anderson III of the United States Court of Appeals for the Eleventh Circuit, in Macon, Georgia. Kaine then joined the Richmond law firm of Little, Parsley & Cluverius, P.C. In 1987, Kaine became a director with the law firm of Mezzullo & McCandlish, P.C. Kaine practiced law in Richmond for 17 years, specializing in fair housing law and representing clients discriminated against on the basis of race or disability. He was a board member of the Virginia chapter of Housing Opportunities Made Equal, which he represented in a landmark redlining discrimination lawsuit against Nationwide Mutual Insurance Co. arising from the company's practices in Richmond. Kaine won a $100.5 million verdict in the case; the judgment was overturned on appeal, and Kaine and his colleagues negotiated a $17.5 million settlement.  Kaine did regular pro bono work. In 1988, Kaine started teaching legal ethics as an adjunct professor at the University of Richmond School of Law. Kaine taught at the University of Richmond for six years, and his students included future Virginia Attorney General Mark Herring. He was a founding member of the Virginia Coalition to End Homelessness.  Kaine had a largely apolitical childhood, but became interested in politics in part due to the influence of his wife's family and his experience attending Richmond city council meetings. In May 1994, Kaine was elected to the city council of the independent city of Richmond, from the City's 2nd District. He defeated incumbent city councilman Benjamin P. A. Warthen by 97 votes. Kaine spent four terms on the council, the latter two as mayor.

Where did he teach?
University of Richmond School of Law.