How do the disguise kit 5e and forgery kit work in dnd?

It does not state explicitly which ability you would use to use tools like disguise kit 5e & Forgery kit. In 5e dnd, It’s up to the DM to decide which capacity will be helpful in any ability check. The default matching of skills and abilities is not a guarantee. This section includes Chapter 7 of the Player’s Basic Rules as well as the PHB. Variant: Skills that have different abilities

Your DM will decide which skill or ability applies to each ability check. That is especially true for tool-based ability tests, as they don’t have default skills. But The two tools ( Disguise kit 5e & Forgery kit) proficiencies are included in the list.

Intelligence checks of d&d Forgery kit 5e & Disguise kit.

Both forging documents and disguising a disguise are covered under the other Intelligence Checks Section of the Ability Checks Guide in the PHB. That would be the default.

For example, this could be history? Nature Or just a generic Intelligence Check (with a DM supplied DC) indicates the need for some special knowledge (remembering the exact race, knowing the names you should include in your document, remembering the handwriting of someone, etc.). It is to create the disguise kit or Forgery kit 5e dnd effectively.

Other checks may be required at the discretion of the DM.

In some instances, you could imagine a DM requiring Dexterity. You can check the Intelligence test in place of or alongside the Intelligence Check. It is to determine if your physical capabilities are sufficient to copy another person’s handwriting convincingly, apply precise makeup, etc. There may be additional checks after creation to ensure it passes. In the description of insanity, one example checks. You can disguise yourself. Once you have created the disguise with the kit (and most likely an intelligence check), you might still need a Deception test to pull it off.