First you need to play the card of the faction you’re currently Closest To, and break your ties to them. Then after you’re broken your ties, you can draw the card of the faction you want to become close with, and play the relevant option on that card.
The exception is Helicon House. You can become affiliated with Rubbery Men or Bohemians directly without the extra step of breaking your current affiliation. This is the only way to become Closest To Rubbery Men, and oddly also the only way to stop being Closest To Rubbery Men (by becoming Closest To Bohemians instead).