How far have you gotten in the fight? Here’s what you’re in for:
Aim for the eyes (as you’ve been doing), and dodge his attacks. His spray shouldn’t be hard to dodge, and he broadcasts his rock attack so you can get moving to stay out of their way. I think the rocks fall in a scattered pattern depending on where you are standing when they’re tossed, so when you see him reaching for a mouthful of dirt, get to one side of the area. When they’re released, move to the other. I’m not 100% on this, but moving as far as you can when they’re released really helps.
After the eyes die, he’ll spawn a queen worm with a bunch of other worms being added. The Leviathan will still attack, but I seem to recall that the attacks were less frequent. Anyway, after you kill one, he’ll spawn in another.
When they’ve both been killed, you’ll go back to regular worm v. Leviathan combat, but the remaining eye is hidden. When it reaches for a mouthful of rocks and tosses them, it will hold its mouth in the air for a moment, revealing a final eye under the mouth, but only for a moment, so you need to be able to get some shot on this eye while dodging the falling rocks.
Try to focus only on the Leviathan eyes (besides the queens), or you’re just sort of stalling for combat - the worms will always be there and you’ll just burn ammo (though that combat can be fun in its own way once you get the hang of it).
Also, weapons with fast travel speed (as in, not an Unkempt Harold, anything Torgue, Twister, Sandhawk… you get the idea) are needed to hit those long-distance shots. You could probably pull these off with +11 in Velocity though. Anyway, there is always the cheese spot that @VaultHunter101 mentioned, though I think at least one of the eyes on the far side of the Leviathan will require you to come out from behind cover.