There are a lot of contributing factors when it comes to eye direction and so it's a tricky thing to nail down. It's not as easy as just placing the eye aim control on an object and calling it done. In my experience the only thing the eye aim is good for is making sure the eyes aren't locked to the head; it is almost guaranteed that your character won't be looking at that control directly. Getting it the rest of the way and making it really appear as though your character is looking at a point in space is a bit more involved. Let's go through some of the things I do to try and nail eye line.
First things first is placing the aim control on the object the character is focusing on. I like having that visual representation of what the character is looking at, so I do my next broad offsets using the individual eye controls. I'll drop a camera on the object in question and offset the eye rotations so they look like they're looking directly at the camera. With this we at least know we have a solid technical base; the character is definitely, technically looking at the object. This is not usually where things stop.
Another thing I do to help with this stage of things is constrain cameras to the character's eyes so you can technically see what they're seeing. This is more interesting than it is helpful. I'll also constrain long, thin cones to the eyes to try and get a feel for what the eyes are looking like from the camera view. The theme you may be noticing at this point is that the technical considerations rarely get you to good eye lines. There are just so many things influencing the perception of eye line that a straight technical approach will not work.
The next thing then is to look at the eyes as a graphic shape, looking at them like a 2d animator. I'll often use my flat shading trick from a previous post to help in this regard. The important things to look for are the amount of white around the iris to the left and right ( top and bottom only really if the eyes are wide open) and the apparent angle of the eye in relation to the head. The eyelids themselves can do a lot to change the perception of that angle.
Don't be afraid to adjust head angle to help your eyeline. I had a shot where the apparent eyeline was too high. In the end, I barely touched the eyes and most of the adjustment came from head angle. You will have situations where that's not possible, like a snooty person looking down their nose, but most of the time there can be some leeway in head angle.
Watch out for cross eyed characters; eyes are never parallel to each other rotationally, they are usually out from each other a bit. You don't want to go too far the opposite direction, but depending on the character this can affect the appeal (think Hei Hei in Moana).
It's important to have a good eyelid setup as this affects apparent eye line as well; as the eye moves around in the socket it pulls at the lids and displaces the flesh around it. The corneal bulge of the eye also deforms the eyelid and should be a part of the setup as any change in the eye area can also, obviously, change the appearance of the eye.
Make sure as you are posing extreme eye positions that you're not digging the eyes in so far that the pupils can't actually see their target. Nudging the eye out, or changing the shape of the eyelid can help in this regard. Similarly with the eyelids keep your pupil clear unless you are deliberately going for a sleepy character.
The shading of the eye also makes a difference; I have often found the eyeline changing ever so slightly once a character is rendered. Having shaders with specular set up in your Maya scene helps give a character that needed spark of life, but refraction also makes a difference. Either you need a clever rigger to make an eye refract deformer (I think there's one in an old Maya book, Maya Hyper-realistic Creature Creation I believe) or else try and get your shot rendered as early as possible.
Get your character's brows involved; subtle adjustments can help support your eye direction. Nothing moves in isolation and the eyes are no exception. As the lids move and deform so too will they affect the brows. This will probably be subtle, but it will reinforce what your eyes are doing and give you that fine detail that sells the performance as real.
Most importantly get other people to look at it, even non-animators; especially non-animators. Granted, people can and will have different opinions about where someone is looking, but it's useful to get those opinions to fine tune and close in on good eye direction. And as always, flip, flop and invert your animation to get your eyes fresh. A mirror of your animation should still be looking at its intended target.
Eye line is a surprisingly tricky and subjective thing. Getting in the ballpark is easy, but that's not usually enough to get that solid connection. Hopefully these tips can help you knock it out of the park
Keep on keyframing!