All you have to do is use a context connection.
At the moment it may not seem obvious, but CLR code always executes in some context. There is always something (somebody) that executed the code directly or in some parent scope (stored procedure calls CLR function, CLR trigger fires on insert etc). When program execution reaches context connection, it just takes connection parameters (server, database, SET options etc) of that scope.
I want this method/storedproc to refer to a specific known table in the database.
After context connection is opened, you are querying your database as usual.
using (SqlConnection conn = new SqlConnection("context connection=true"))
{
conn.Open();
SqlCommand cmd = new SqlCommand("SELECT * FROM MyTable");
SqlContext.Pipe.ExecuteAndSend(cmd);
}