The phrase "Run the table" comes from the game of Pool, or Billiards.
In a popular game that uses a pool table, a player tries, with one shot, to hit a ball into a hole on the side of the table.
If he suceeds, he can try to hit another one before the other player has his first turn. He can continue to keep playing until he misses one. Then the other player can take a turn... and he plays until he misses.
If a player is good and suceeds, and suceeds, and suceeds, and suceeds, until he has hit every single ball off of the table, while the other player has to just stand and wait without every getting a chance to play, he is said to have "run the table" Since he never missed a single shot, the other player never got a chance to play.
So in your examples, the person has to "start winning and win every single time; if he makes a mistake and loses even one thing, the other person will have a chance to play and beat us."
If you have to do BETTER than win a lot, but you have to WIN EVERY SINGLE TIME, then you can say that "I need to run the table."
Use this sentence when you (or someone) has to be perfect, or you will fail. Do you need to score a 100 on an English test? Then you better be able to run the table... one mistake, and you will not get a 100.