Coulter was raised in St. Clair Shores, Michigan, and attended St. Joan of Arc Grade School in the city.

In 2007, Coulter completed Harvard University's John F. Kennedy School of Government program for Senior Executives in State and Local Government as a David Bohnett Foundation LGBTQ Victory Institute Leadership Fellow.

[9][10] In 2013, Coulter, along with Congressman Sander Levin, undertook a challenge in which they spent less than $4.50 a day on food.

Coulter ran for a full term in the position in the November 2020 election,[13] and defeated county treasurer Andy Meisner in the Democratic primary.

Coulter won a second full term as county executive, defeating Republican businessman Nikola Gjonaj in the November 5, 2024 general election.