Can I grapple the enemy and then use him for cover?
RAW: If it blocks at least half your body, it gives you cover
As long as something blocks your body, you can benefit from cover.
Let me show you the Grappler's Manual. Here's the relevant section from that guide.
Human shields: If an enemy is blocking at least half of your body, you'll get +2 to AC and Dexterity saving throws from half cover (PHB, 196). If you're grappling enemies, you have at least one, possibly two, enemies who will be more than happy to stand between you and danger.
Here's the rules for cover cited above.
A target with half cover has a +2 bonus to AC and Dexterity saving throws. A target has half cover if an obstacle blocks at least half of its body. The obstacle might be a low wall, a large piece o f furniture, a narrow tree trunk, or a creature, whether that creature is an enemy or a friend.
Note that you might even be able to get three-quarters cover this way.
But, with a caveat...
There is no direct rule saying that you may grapple someone and use them as a shield. So you will have to very deliberately make the cover rules relevant, by justifying that at least half of your body is blocked by the grappled creature.
Additionally, note that the cover they grant will probably be only towards a certain direction or angle. They may just be able to step to one side and your cover benefits are nullified.
Finally, cover applies for all kinds of attacks because it adds to AC. So even melee attacks are affected by this, and it stacks with a shield. This might be a concern for the DM. But perhaps, to think of it more reasonably, if you're on a grid and someone is trying to use a polearm to hit you but through the space of the grappled creature, it becomes much easier to accept that you can gain the cover bonus.
You can definitely use an enemy creature as cover, and you don't even need to grapple them. The rules for cover mention using creatures as cover several times, and in one case, specifically hostile creatures (SRD p96):
Walls, trees, creatures, and other obstacles can provide cover during combat, making a target more difficult to harm.
For example, if a target is behind a creature that gives half cover and a tree trunk that gives three-quarters cover, the target has three-quarters cover.
The obstacle might be a low wall, a large piece of furniture, a narrow tree trunk, or a creature, whether that creature is an enemy or a friend.