TFT's match history can be looked up through the client or third-party websites.
TFT Match History Lookup: How to Look Up Your History
It has been almost two years since Riot Games added a match history for TFT. Yet most fans are aware that they are rather unkind with providing information.
Unlike the match history for League of Legends games, the ones for TFT only show the player's deck and the final ranking. Games placed higher than third place get a special lining next to that match, but what is the point of that?
It is more efficient for TFT players to rely on outside websites for match history. Websites show every player from that game and even how long each player lasted. Some even show a summarized statistics and what traits have been played best for the person. These websites can be easily found by searching online.
It is hard to believe that almost two years into the first update, Riot Games has not updated the match history system. Players can only hope that there will be improvements with the client in the near future.