I'm a guitarist, but I also play a bit of drums and piano. Not being the greatest drummer or piano player, I totally understand the fundamental problem of coordination between limbs when playing these instruments, particularly when drumming!
Firstly, it has to be said, the most important thing to do is to practise slowly; only by playing a pattern carefully, correctly and consistently will you be then able to play it well at a faster speed.
Secondly, I'm going to offer similar advice as I would to a guitar pupil, that is finding it difficult to focus upon what both hands are doing at the same time. Two things:
simplify the problem. If you can play small parts of the overall pattern correctly, you'll be able to combine these to then play the whole pattern.
treat what you're doing as a purely mechanical exercise. Yes, it's very important to play any music, on any instrument with musicality and expression, but initially you just need to get your hands and feet moving where you want them to! Once these movements start to become comfortable, relaxed, intuitive it will then be easy to think about expression and musicality.
So, how would you apply all these ideas to the pattern you describe above? This pattern is certainly a good place to start; when doing group band sessions with drum teachers, it always seems to be the beat they try to teach non-drummers to play first (and it is easy to hear this pattern as being the basis of a lot of rock and pop drum parts).
I would try two ways to build up your coordination, in order to play this pattern. These approach the problem from two different angles; trying to learn something like this often seems easier to me, if you try it in a couple of different, complimentary ways.
Essentially, this pattern only has three different "events": 1. hi-hat on its own; 2. hi-hat and kick drum; 3. hi-hat and snare. First of all, make sure you can easily play each of these. This seems like such a basic step, that it might seem unnecessary, but ensuring you can do it, will make playing the whole pattern much easier. Then, work out where each of these "events" occur in the pattern you are trying to learn. Luckily, for this pattern, these occur with a regular rhythmic pattern, on every quaver (8th) of the bar. Using the numbers above, the pattern would be: 2 1 3 1 2 1 3 1 If you play this succession of combinations slowly, you should then be able to gradually increase the speed, and as you do so you should start to feel that the coordination becomes automatic, that you can do it "without thinking" (or without thinking as much!)
Gradually build up the pattern instrument by instrument (or limb by limb, if you like). The hi-hat quavers run throughout this pattern; firstly ensure you are comfortable playing these evenly in a relaxed manner, then simply add the other parts of the pattern in, one-at-a-time! You could either add them in in order, first the kick on beat 1, then snare on beat 2 etc, or you could add each of the kicks, then add the snares one at a time. Again, it's important that you do this slowly, at a speed that allows you to play accurately. And, it is important to repeat the pattern with the added note multiple times, before adding in another note. This allows you to confidently learn the whole pattern.
Finally, this is the perspective of "a-guitarist-that-plays-drums-a-bit", it'll be interesting to see how drummers reply to your question!
And, most importantly of all - use a metronome!