It's been a while since I've tried this syntax... but in SQL Server you can specify a from on an update.
UPDATE Emp SET
UserName = Left(FirstName,1)+LastName
FROM Emp e1
WHERE NOT EXISTS (
SELECT *
FROM Emp e2
WHERE e2.UserName=Left(e1.FirstName,1)+e1.LastName
)
EDIT: My syntax certainly runs but I'm not certain that it's correct. Regardless of whether or not it's right, I would suggest using the alias in the update statement just to ensure that others can better understand what you are doing.
UPDATE e1 SET
...
FROM Emp e1
...