How closely is this game tied to the TV show?

Playing this game isn't necessary at all. The game's story doesn't affect the story in the tv show or books. It follows a smaller house/family in the north, House Forrester (originally loyal to the Starks, now in turmoil) that is never mentioned in the show and barely mentioned in the books. Though some of the new characters in the video game interact with a few characters that are in the show, it's not anything that changes the trajectory of those characters in the show. The events of the game take place between seasons 3 and 4 of the tv series.

The show's writers have supposedly said that some of the characters in the game will eventually appear in the show, but as far as I recall that hasn't happened yet, and when (or if) it does happen, I highly doubt it'd be in a way that would require viewers to play the game to understand what is going on.

But it is worth playing imo, if you enjoy the storytelling in game of thrones. I found the characters really interesting. Playing it made me wish the tv show followed this family as well, especially with the weird directions they've been taking the storylines in the show in the latest season or so. The game reminds me of the first few seasons, which had really great character and world-building.

It might be a good thing to mention also that the disconnected relationship between the game and show doesn't work the reverse way quite as well. The game assumes that the player has seen the show and knows what's been going on with the major houses, especially with the Starks. It's probably still playable without having seen the show or read the books, but the game doesn't offer much exposition to anything that would have been learned in the show.