I assume this is a browser based application. The standard is cookie authentication. The user data (claims) are cached in the cookie. The cookie authentication middleware reads the cookie content on each request and creates a user principal which drives the standard Core security for the current request.
To get the user is simply...
User.Identity.Name
Use cookie authentication without ASP.NET Core Identity
If I can reiterate, what is the best practice for creating a relationship between the logged in user and the tables that store this particular users health information?
You have not explained how the security works so it is hard to provide an accurate answer. Usually the user's Id is used to identify user records. Each record that belongs to the user will contain the user's Id. When the user is logged, they can only see records that belong to them. The query is simple, fetch all the records where the Id = the user's Id.