The error message has nothing to do with what you want to do. Its related to your method. The message is telling you that the field
$Action.accountId.New doesn't exist on your Contact Object. This makes sense in that you can't create a new accountId on a Contact. You can relate an AccountId to a Contact, or more accurately a Contact to an Account (contact having a look-up to the Account or being a child of it), but the Account has to first exist.
I have some more comments on what you want to do/are doing that you may want to think about. I don't know what you're use case is or if this is something you're simply "playing" with in a Dev Org. as an exercise. Salesforce is primarily a B2B platform and sometimes used for B2C (individual customer). When a User creates a Lead, it's to create a "prospect" for which a salesman in his Org might be able to sell something to another someone at another business, who for obvious reasons, presumably doesn't work for his company/Org. For that reason, it makes no sense to me that you'd want to assign a User's Company Name to the Lead, even if the Lead is an individual (B2C) sale prospect.
Similarly, in doing what you're attempting with Contacts, you're essentially creating contacts that would presumably all work for your Org/Company. Likewise, these contacts would all be assigned to the same company Account. Even if they do work for the same (your) company, most would have their own unique employee ID or company account number of some kind. As I said, I don't at all know what you're doing, so it's not for me to say that it's "wrong" to do it or that there's no use case for it.
That said, be aware that Users are created from Contacts, especially when it comes to customer community users. All Users have a ContactId. The real info you're looking for that you can count on for accuracy would come from a User's Contact info, not from their User Profile. So if there's a related AccountId, you'd find it in their Contact info; something which you can't get directly from User without running a query using a custom controller.
For additional info you may want to look at these relationships closer in Workbench and also refer to the Standard Objects section of the Object Reference to the various fields that exist in each of them.