Melt chocolate in a microwave-safe glass or ceramic bowl in 30-second intervals, stirring after each melting, for 1 to 3 minutes. Cool melted chocolate to room temperature, about 5 minutes.
Beat butter in a bowl using an electric mixer until light and fluffy; add cocoa powder and beat until well combined. Mix 1 tablespoon coffee into creamed butter mixture; add melted chocolate and mix well.
Beat confectioners' sugar, 1 cup at a time, into creamed butter mixture, mixing well before each addition until frosting is smooth. Add remaining coffee, 1 tablespoon at a time, until desired consistency is reached.