Consideration Of Tactical Utility
Firstly, consider why this is useful: a grappler-based character is probably not going to be doing a lot of damage. Grappling is more about battlefield control and shutting down a single target. A grappler's primary strategy is to hold the enemy, while the rest of the party hits them. That's not to say they can't pick up a sword when they don't feel like grappling, of course.
That said, 1d6 damage is quite more than a 1d2 unarmed strike if the grappler is unarmed. Also note jumping is movement, not an action, so you still get to attack when you're done.
To get at least 10' of jump height (dealing damage and knocking them prone), we can play with different Strength scores and heights.
Below are the minimum calculations, which a character can do right from level 1.
Minimum Stats At Str 10
If you do a standing high jump, you get half of 3 + Str Mod. Assuming a Str of 10, a standing high jump can reach 1.5'. This leaves 8.5' left that we must gain by a character's reach. A character's reach is 1.5 * height: 8.5' / 1.5 = 5'8"
Amusingly, a lanky 5'8" human with Str 10 can already pull this off. They would need to use only 6' of movement, too: 1.5' (jump height) * 2 (for jumping) * 2 (for grappling) = 6'.
So the minimum is:
Strength: 10, Movement: 6', Height: 5'8"
With a higher Strength score, you can afford to be shorter. You don't have to, though. You can break up your movement. If you're at least 5'8", you could theoretically spam powerbomb suplexes by spending all your movement doing this, but jumping only 2' of the time, eating 6' of movement per try. If you have 30' of movement, you can do five suplexes -- 5d6 for movement. This sounds a little ridiculous though.
Anyway, with higher strength means you can afford to be shorter by jumping higher. Below would be the min stats for higher values of Strength.
Minimum Stats At Str 12
Standing high jump height: 0.5 * (3' + 1') = 2'
Minimum Needed Height: (10' - 2') / 1.5 = 5'4"
Minimum Movement Needed: 2' * 2 (for jumping) * 2 (for grappling) = 8'
A short human can pull this off.
Minimum Stats At Str 14
Standing high jump height: 0.5 * (3' + 2') = 2.5'
Minimum Needed Height: (10' - 2.5') / 1.5 = 5'
Minimum Movement Needed: 2.5' * 2 (for jumping) * 2 (for grappling) = 10'
A tall dwarf with Str 14 could do this.
Minimum Stats At Str 16
Standing high jump height: 0.5 * (3' + 3') = 3'
Minimum Needed Height: (10' - 3') / 1.5 = 4'8"
Minimum Movement Needed: 3' * 2 (for jumping) * 2 (for grappling) = 12'
A strong, average-height dwarf can do this
Below, I try to maximize the damage through this strategy with a playable character. This is no longer useful/sustainable, so it's just for fun now.
Choosing Race For An 8' Height
A tall goliath will be 8'. With the Enlarge spell, they're 16' tall. A high jump lets them add 1.5 * 16' = 24' to their height. Their base speed is 30'.
Strength of 24
A level 20 Barbarian will have Str 24, move speed +10'. A standing long jump will be 0.5 * (3' + 7') = 5'. This requires 5' * 2 * 2 = 20' of movement. You also have +10' of movement, total 40'.
Maximizing Jump Height With Spells, Feats, And Items
A running high jump will get up to 10' but it will need 20' of movement for a running start (10' * 2 for the grapple) on top of that, for 60' total (10' base * 2 jump * 2 grapple, plus the 20' running start).
With the Athlete feat to halve your running long jump distance, you only need 50' of movement for the whole jump. With the Mobile feat, you get +10' of movement for a total of 50'. The Longstrider spell can also give you +10' for a total of 60'.
With the Jump spell, you get thrice your jump height -- 30' from 10'. With the Boots of Striding and Springing stacking with Jump, you can jump up to 90' if you have that much movement. This is because 10' base jump * 3 Jump spell * 3 Boots = 90'.
With the Haste spell, your max jump height is at least not cancelled out by grappling someone while jumping. So your 60' of base movement is doubled to 120', but the grapple reduces it back down to 60'. So you can jump up to 60' while grappling someone.
Jumping And Dashing
Your initial jump movement would bring you up to 50' (effectively 60' movement, minus the initial 10' running start). Assuming you start the turn already having grappled the target, you can take a Dash on your regular action, an extra 60' of movement. Also, with the Haste spell, you can Dash -- affording you an extra 60' jump height.
In total, with spells and magic boots, you can jump up to 170' (movement plus two dashes).
At the DM's discretion, you can try to jump an extra 6' with a successful Strength (Athletics) check. If successful, your total jump is 176'.
Your Enlarged Goliath adds 24' to his reach, and you can jump 176' due to magic.
You can bring the opponent up to 200' with the help of spells, with yourself having jumped 176'. Target takes 20d6 maximum fall damage and ends prone. You take 17d6 fall damage and fall prone, too. The Athlete feat lets you stand up with only 5 feet of movement instead of half your movement, though. A teammate can cast Feather Fall on you to prevent damage.
Without spells, you might be able to bring the target up to 50' (10' jump + 24' reach + 6' athletics check + 10' dash), with yourself having jumped 26'. Your target gets 5d6 damage, falls prone. You get 2d6 damage, fall prone as well.
Strength: Level 20 Barbarian. Str 24
Movement: 100' (Mobile feat, Haste and Longstrider, Barbarian's Fast Movement)
Height: 8' Goliath
Spells: Jump, Longstrider, Haste, Enlarge, Feather Fall
Items: Boots of Striding and Springing
With Spells: 19d6 or 20d6 max fall damage with a nice DM; you take 17d6 (or none with Feather Fall)
Without Spells: 4d6 or 5d6 with a nice DM; you take 2d6