What you roll exactly is going to be usually a social skill check.
Which one will be a DM call, depending on what you are trying to do. But there are some non-magical "hard" mechanics you can use.
Ask the DM if you can use a skill feat from Unearthed Arcana called "Diplomat".
It grants +1 charisma, proficiency or double proficiency in persuasion, and the ability to charm someone who isn't in combat with you (or your allies) by talking to them for 1 minute and beating them in a skill contest (until they are more than a certain distance from you, and 1 minute longer).
Charming in 5e means they cannot attack you (yawn in a social context, but has some fun to it; a guard who you Charm cannot use violence to prevent you from doing something), and you gain advantage in further social skill checks.
Another good one is Empathic, where you can burn an action, make a check, then gain advantage on your next attack or check on the target. It gives you insight proficiency (or double proficiency if you already have it) and +1 charisma as well.
As written, these stack; the round before you make your Charm check, you can Empathic (Insight vs Deception), and if successful get advantage on the (Persuasion vs Insight) check. It even fits thematically.
Note that the Charmed condition is not the Charm spell. The person Charmed does not know they are under the condition; the spell, in comparison, informs the target.
This is a lot of work to apply the Charmed condition, especially because you must succeed at a skill check to apply it. But if the DM is likely to ask for more checks, or different social skills against different targets, it can help. And, as written, it cannot hurt.