I will get everything you need to know about the story out of the way right now because it's pretty much nonexistent as was the case in many SNES era side scrollers. Fat Alligator stole your bananas, so know you are going to kick him in the dick (Or jump on his head several times). While the game's animations do a great job of giving some personality to characters who don't talk, do not come into this game expecting any kind of actual story. This is old school platforming action and nothing more.
With any kind of platformer, the controls must handle well. In the case of Donkey Kong Country, I think it handles wonderfully. It may not be quite on the level of something like Super Mario World, but it never feels like there is anything wrong with the game collision detection. At worst certain segments where timing is involved, though only a few of them, can feel a bit awkward. I'm getting this little bit out of the way now because given the game only has the gameplay to offer you, if the controls were poor it would kill the entire game. Especially since you can die at a level relatively easily if you slip up.
On that note, the game is an example of a game that isn't that difficult compared to most others, but at the same time does offer up a challenge. One hit will eliminate whatever character is the one being controlled, either Donkey Kong or Diddy Kong, and switch over to the other. You can recover the other by finding a DK barrel and breaking it, though you lose if both Kongs are gone. There are generally only a couple barrels in each stage, and falls into a pit will result in instant death. There is an odd delay when one Kong goes as well, so depending on where you get hit it's possible protecting the second isn't an option. The game is pretty demanding in terms of forcing you to pay attention to what's going on.
At the same time so long as you are keeping on your toes, nothing in this game is particularly challenging compared to other games of the day. All of the challenges in the game are pretty well designed as a sort of entry-level game for people who just aren't that experienced at gaming, and it does a good job of this without any kind of excessive hand-holding.
And even without the use of tutorials, the game does a really good job of showing you how everything works. Early on it does this by showing you some bananas you can collect in a downward arrow pattern, and if you jump up the tree and follow it, you break open the ground and find stuff. Know you know doing this is an option, and similar things happen concerning breaking open walls as well as tips from visiting Cranky Kong in various areas. Right away the game gives you a lot of visual clues at how to discover the various secrets along the way, so if you pay attention there are a lot of different things to find.
That said, as many old games go the reward for doing all of this is essentially building up more and more lives so that in the later levels you'll have an easier time of clearing stages. If you die in a stage, so long as you didn't lose your last life, you can start the level at the halfway point so long as you reached it. Other than that finding secrets in this game is doing so just for the sake of it. And once you start reaching the end game, its easier to just get lives by going back to the games very first stage.
That said, so long as you like playing through the levels just for the sake of enjoying the game, you get a good variety of stages. Each world has it's own themes and quirks, and every stage in each world uses those aspects in different ways. It's a very short game, which is a shame, but there isn't a part of it that feels dull or repetitive. It knows how often to change things up, all while keeping the core of the game the same throughout.
As mentioned at the beginning, even with no real story to speak of the character animations are so well done their character still shines through. It's obvious who Donkey, Diddy, and even Cranky Kong are. And for as little time as you actually see King K. Rool, even he is rather expressive. It's easy to see how people grew to like this cast of characters despite the lack of story, I kind of wish more games put so much effort into their visuals these days.
I don't have anything else to say about this game. It formed the core of what would be used n later games of the series, and it was just a strong foundation to build on. If old school side scrollers are your thing, or if you're just interested in experiencing part of what many consider the best era of the genre, give it a go.
Donkey Kong actually can be a hard game at times, when you first play though. It's also the stages needing quick reactions and thinking to avoid dying that make the challenges even better.
Diddy's Kong Quest was always my favourite, then this one just because of how much better it was and how much I loved it as a kid (and still do now), but Donkey Kong 1 is well worthy of the play too.
So many quality characters, amazing sound and music and great games overall.