Two basic methods: increase statistics, increase the number of attacks
One of the features of D & D 5th is that weapons have fixed damage dice unless they possess versatile property: their one-hand or two-handed utility. The claw does not fall into this category as would a long sword (1d8 / 1d10) or a personal quarter (1d6 / 1d8).
Unarmed strikes also have fixed damage: a tabaxi claw attack is an unarmed strike.
If you hit them, you deal sharp damage equivalent to 1d4 plus your
force modifier, instead of normal blunt damage for a
unarmed strike. (Volo Monsters Guide, 115)
Without attack bonus, the claw is 1d4. (Average 2.5).
With a strength of 18, the claw is 1d4 + 4. (average 6.5)
This is an increase of 160%.
Compare this to a normal unarmed strike: 1 + Str mod.
On one stroke, an unarmed strike deals blunt damage equal to 1 + your Force modifier. You control your unarmed strikes (Basic Rules, page 76)
In short, tabaxi already has "more damage" with claw attacks.
With more attacks, the claw is 2d4 (5), 3d4 (7.5), 4d4 (10), depending on how you set up your computer for additional attacks: as a Fighter, Ranger, Monk, etc. Depending on the strength or dexterity applied to the attack, add the bonus to the damage dealt.
An additional way: to score a critical hit
This will add an extra d4 to the damage.
If the tabaxi wants to do more damage (in terms of larger damage) per hit, he must use a weapon like any other humanoid race (or become a monk and gain levels in that class).
To be a monk
You can use dexterity instead of strength for attack and damage
rolls of your unarmed strikes and monk weapons.
The choice of Monk's character class will ensure that the unarmed strikes are 1d4, like the Tabaxi attack; damage is increased for unarmed strike damage at 1d6 at level 5, 1d8 at level 11 and 1d10 at level 17. Technically, this damage is sharp, but you may be able to work with your DM to reduce it accordingly. the type of damage of Tabaxi.
Magic object: claw badge
This article appeared in the Horde of the dragon queen published adventure. It adds +1 to hit and +1 to damage inflicted on both unarmed attacks and natural weapons (Tabaxi's claws meet both criteria). Whether or not it appears in your campaign depends on your DM.
The jewels of this dragon cult candlestick badge with a purple light as you enter into combat, reinforce your natural fists or natural weapons. When you wear the badge, you gain a +1 bonus for attack rolls and damage rolls that you suffer with both hand and natural weapons attacks. Such attacks are considered magical.