Do the claws of Tabaxi and Tortles change the unarmed attacks of monk to slashing damage, or is the damage separate? Does the unarmed slashing damage increase with the monk's level, as described on the monk table in the PHB?
Both tabaxi and tortles use the following phrasing:
Your claws are natural weapons, which you can use to make unarmed strikes. If you hit with them, you deal slashing damage equal to 1d4 + your Strength modifier, instead of the bludgeoning damage normal for an unarmed strike.
Note that you can use your claws to make unarmed strikes; this means that if you want to, you still have the option of making normal unarmed strikes that deal 1 + your Strength modifier bludgeoning damage.
Now, the Monk:
You can use Dexterity instead of Strength for the attack and damage rolls of your unarmed strikes and monk weapons.
You can roll a d4 in place of the normal damage of your unarmed strike or monk weapon. This die changes as you gain monk levels, as shown in the Martial Arts column of the Monk table.
The first bullet point is simple enough. You can use Dexterity instead of Strength for unarmed strikes. Note that, again, you can - you don't have to, so you can still choose to use Strength if you prefer.
The second bullet point says that you can roll a d4 in place of the normal damage of your unarmed strike. The question here is whether the "normal" damage of your unarmed strike is the default 1 + Str bludgeoning, or the 1d4 + Str slashing that your claws do. Ultimately, this will be up to your DM to decide. Personally, I'd say that your unarmed strikes "normally" deal 1d4 + Str slashing, so you should be able to replace the die with the Monk die. Importantly, it only replaces the die, so if your DM rules this way, you can still deal either slashing or bludgeoning as you choose.
The monk's Martial Arts feature says (PHB 78, emphasis mine):
You can roll a d4 in place of the normal damage of your unarmed strike or monk weapon.
Both the racial trait and the class feature provide a way to calculate your unarmed damage. They do not add to or modify, but replace it. As such you have to choose which one to use and cannot combine them. They are also separate features and have no effect on each other.