To defeat Akuji, make sure that, before you start the cinematic to battle him, you first clear the room below you of any Ronin resistance. Once the match begins, move to the stairs that you walk up before to get to that deck. Begin walking back down and make sure Akuji is falling you as you go. As you go back down stairs, there will be a location on the ground scorching with fire. Make sure you are a decent distance around the fire and try to make Akuji catch on fire. This depletes much of his health. You may only need to do this about three times to kill him.
To kill Akuji in the tea house, use L2 to counter his attacks. It inflicts damage and knocks him down. When he is down, stab him twice rapidly before he gets up. Repeat this until he has been killed.