First off, you will want to take the Grappler feat at some point. However, it's worth noting that the primary benefit of the Grappler feat is getting advantage on attacks against a target you have grappled. Because your build is built around grappling, this means that you'll basically have advantage on most attacks. The part of that feat that allows you to restrain the enemy is less powerful - because you are also restrained, you give up your advantage on attacks, and leave yourself vulnerable to attacks from other enemies.
To focus on grappling, you'll primarily want to focus on improving your Athletics rolls as much as possible. A few routes to do so:
- Take one level in Rogue or three levels in Bard to get Expertise in Athletics. This will give you an additional +2 to +6 to your rolls, scaling up as you level.
- Take a few levels in Barbarian. Either just one level, or take three for the primal path features and an additional rage. While raging you would have advantage in Athletics checks
- Raise your strength score as much as possible through ability score increases and magic items
If you do all that, it's perfectly possible to have advantage and a +17 to Athletics checks at level 20, which means that basically nothing will ever escape you.
Now that we've made you inescapable, how do we make grappling effective? I think you already made the right choice with monk - unarmed strikes are still just as effective when you're using a hand to grapple someone, and it's a thematic fit. Any monk will work, but Open Hand gives you a cool combo:
Take the attack action, and grapple the target. As a bonus action, use Flurry of blows to knock the target prone. Per the rules on being prone, you can't stand up if your speed is zero. This means that they have to break your grapple to stand up, which will be incredibly difficult to do. While prone all of their attacks have disadvantage, and anyone in melee range has advantage on attacks against them. You pin them down, and your party can put the boot in.