My monk character has the Grappler feat and currently wears a Cloak of Arachnida. Can I move a Grappled character up the wall with me? Do I have to make a STR check?

I'd argue no -- and would rule that way if I was running. But my DM disagrees. (We're trying to figure out how to adjudicate this, not in game right now.)

The cloak gives you a hands free climb speed. I'd say there's no additional check for moving a grappled creature, just the normal grappled/movement rules.

Considering you can carry a Grappled creature even while jumping, I'd say it's fine to grapple a creature with you in that scenario, especially since you don't even need to use your hands for climbing.

As mentioned by AndredSkratch, your movement could be affected by the Grappled rules depending on the target size, and by the Climbing rules (Cloak of Arachnida gives you Spider Climb, though).

A couple of related threads:

• Climbing while grappling
• Jumping while grappling

EDIT: for clarity.

The default grappling rules are somewhat strange since they allow you to carry a creature that is larger than you even when you might already be very close to the limit for your Carrying Capacity (based on weight carried and your Strength score).

I think that a very reasonable house rule would be for a DM to set a default value for the weight of a creature of each size and then factor in that added weight to see if you are able to move while using the carry/lift/drag rules to force the grappled creature to move along with you while maintaining the grapple.  But RAW, the grappled creature is considered to be weightless for these purposes.

Here’s the grapple feat 2024:

https://www.dndbeyond.com/feats/1789147-grappler

As long as the creature or person is your size or smaller, no penalty for movement when you move with the grappled person/creature.

From cloak,

Spider Climb. You have a Climb Speed equal to your Speed and can move up, down, and across vertical surfaces and along ceilings, while leaving your hands free.

So, the following will work its RAW and RAI together.

You make an unarmed strike attack (During your Action- not BA) 

You apply damage and grapple the opponent. (If applicable, make additional attacks if you are 5th level, haste etc)

Then BA (step of the wind) dash double your movement. Drag and climb up with whatever you have grappled. Then simply let go causing 1d6 for every 10 ft (mav 20d6). and sit back and watch the splatter. 
Slippers of spider climb will work same way.

But you can rule it however you want, but that’s written and intended in a nutshell.

Thanks everyone!