What would be the best way to structure and mine this set of data?
How best could I organize this data, in what type of database?


A simple relational database should do, but you could also use a "fancy" graph database if you want. One table for the users, and one for the "interactions". Each interaction would have foreign key columns for the two participants, labeled winner and loser, and the number of the episode the interaction it occurred.


  Also any ideas on the best way to visually represent this data?


A graphical representation for social network analysis suggests itself. Here are some papers and a subreddit for inspiration. In your case, there is a concept of competition with clear winners/losers, so you could make your graph directed. Have the characters be the nodes, and add directed edges from the winning party to the losing party for each interaction. Collapse repeated interactions, etc. This approach would let you quickly identify overall winners and losers, as well as simply who interacts with whom.