Why was the character Coldhands not introduced in Season 4, when he was in the books?

There can be many reasons a character isn't introduced. While I can't seem to find any interviews that directly explain why this character was left out, that could be because they don't want to spoil anything for the remaining seasons.

While I haven't read the books, I watch the show with a few people who have, so I'm aware of several moments in season 4 that did not go according to the book.

The thing I tell them is the same thing I tell everyone who's read the book a film is adapted from: the show writers are telling their story, the book writer is telling a different one. Sometimes they're 100% accurate, sometimes only 50%, and sometimes they go a completely different direction. The reason for changes can be numerous.

1. Things get cut due to budget/time constraints. While TV shows have more time to play with than movies, you have to remember that both need to get through their stories in a set amount of time. 10 episodes per season is pretty short compared to most shows, but most shows don't have the story or budget that Game of Thrones does. As such, characters or events or entire story lines may be scrapped for the sake of time and monetary concerns. Given the amount of characters and stories the books wade through, this is especially possible for a show like Game of Thrones.
2. Writers change things to keep those who've read the source material on their toes as well. One thing about people who've read the books is that there's likely little that could surprise them when they see it adapted, verbatim, to film or television. As such the writers might change things to provide those viewers with their own surprises. Sometimes they work, other times they don't.
3. The writer of the source material is still writing some part of the source material. As it stands, five of the planned seven books are out for A Song of Ice and Fire. Next year we'll see season five start, and if we're lucky book six will be out. However, it's possible that not all the books will be written and the stories completed before the show is ready to finish. Currently, Game of Thrones has been approved for a fifth and sixth season, and that could be that, but we don't know. Martin still has yet to produce the final two books of the series, though evidently the sixth is coming fairly soon. Given these factors, story considerations may need to be made for the sake of the show and the current state of the source material. Evidently George R.R. Martin has told the series writers where he plans for the various stories to end up, so they can sort of plan around that. However, you should probably expect the show and the books to take slightly different paths.
4. Sometimes writers just don't like characters or see the point in them. Unfortunately, you get people writing adaptations who simply don't care for certain characters, or can't figure out how to make them work within the constraints they're playing with (see point 1). A good example is the third Spider-Man film. Rami did not like the character of Venom, and had originally wrote the script without him. However, Sony forced him to include the character citing fan demand. It eventually led to a rift between Rami and Sony, one they couldn't overcome, and eventually the franchise was rebooted anyway.
5. We still have two guaranteed seasons to go, so they could show up later. Just because a character hasn't shown up, "When he's supposed to," doesn't mean they never will. Remember that the show writers are telling their story based on the source material, and may introduce things at differing points from the source material.

Unfortunately, without any official word from the show's writers, anything discussed is pretty much conjecture. I'm simply seeking to provide reasons for why certain things may not be included in film and television adaptations of written works.

Since you are addressing Cold Hands specifically, and his absence, I'll float this (which is built from a comment I made on another, slightly overlapping question), which I think needs to be considered in its own right.

The reason Cold Hands isn't in the TV show is because...

Cold Hands isn't important

Before all his über-supporters jump on me, let me explain myself:

GRRM has been involved with the Game of Thrones TV show from its inception, and has obviously been advising the script department on the significance of certain elements, so they can draw particular attention to them. Much like one on the comments made about Harry Potter from @MattD above, this is to help with character motivation and overall focus.

If he was an important character to the Grand Narrative of Game of Thrones, wouldn't GRRM have insisted on his presence?

Think about it: he has had every opportunity to declare to the Showrunners that Cold Hands is the future king of Westeros, or has some significant part to play that couldn't be substituted by another character. If he was integral to its story, he would be included.

That isn't in any way to diminish his presence in the books, its just to say that he isn't required for the TV show. It functions perfectly without him. The only people who are concerned about his absence are people who perceive the show para-textually alongside the books: try it, ask anyone who hasn't read the books if they think its confusing or lacking without Cold Hands, and watch as they stare back at you with confusion.

Some fat has to be trimmed, and GRRM was obviously happy with them losing Cold Hands: he was in collusion with the show, and as such an accessory to his removal. He can't think much of the character, or at least doesn't consider him to be important...

There are several theories among book readers as to who Coldhands may actually be.

Let's assume Coldhands is actually just another form of the 3-eyed crow/Bloodraven

If this theory is true then it wouldn't be much of a point showing him in the series as Bran is already being led by this being and helped by the Reeds to the 3-eyed crow's lair.

If the theory that he is a member of the Night's Watch is indeed true I can then also see why this would be not necessary considering of what they showed of Bran's journey seemed to be just a struggle through the North to the great Weirwood tree.

However, one theory also states that Coldhands is actually The Night's King.

Which is who this guy is rumored to be.

Whether or not this is the actual Night's King is still to be determined. It is also still to be determined who this guy actually is, meaning it is quite possible that this could be Coldhands.

As far as why, no one is really at liberty to say except the show-runners themselves, unfortunately. And from what I can find there hasn't been much of a word on Coldhands appearing or not appearing.

All we have is our theories.

Pure conjecture on my part, haven't read the books, but extrapolating from some other answers, Could it be possible that

1. Coldhands wasnt that important
2. coldhands is Benjen Stark
3. the actor who portayed Benjen was unavailable

If you have this character, Coldhands, who isn't really important other than the fact that he resolves what happened to Uncle Benjen, and the actor who portayed Benjen is not available (for whatever reason) it makes sense to cut the character entirely rather than recast. But I'm literally just making this up as a plausible scenario and have absolutely no evidence. IMO most likely Coldhands either isnt important enough, or as was mentioned, he will make an appearance later on, when his presence is more crucial.