Update July 2020
A new feature has been added to the Copy activity recently to allow you add columns, $$FILEPATH is currently the only supported variable. See here for more detail:
https://docs.microsoft.com/en-us/azure/data-factory/copy-activity-overview#add-additional-columns-during-copy
Original Answer
Adding an extra column to a dataset might be considered Transform and the Azure Data Factory v2 (ADF v2) Copy Task does not lend itself easily to Transform. It can do a couple of things like convert from one format (eg csv) to other formats (eg JSON) but it is limited. Maybe at some point in the future they add something to the mapping which allows adding string literals or something similar to the SSIS Derived Column feature, but these type of features are getting added to Mapping Data Flows at the moment it seems.
One way to achieve this however, is to use a stored procedure target with a parameter for the filename and a table-type parameter for the main dataset. It looks a bit like this:

The downside is you now have to create a supporting table-type in your database (CREATE TYPE) and a stored proc to handle it, something like this:
CREATE TYPE dbo.typ_multiFile AS TABLE (
col1 CHAR(1) NOT NULL,
col2 CHAR(1) NOT NULL,
col3 CHAR(1) NOT NULL
)
GO
CREATE OR ALTER PROC dbo.usp_ins_myTable (
@fileName AS VARCHAR (100),
@typ AS dbo.typ_multiFile READONLY
)
AS
SET NOCOUNT ON
INSERT INTO dbo.myTable ( [fileName], col1, col2, col3 )
SELECT @fileName, col1, col2, col3
FROM @typ
RETURN
GO
Note the Copy Task is inside a ForEach task, as per this diagram:
