SQL Server 2012 – How to Remove Duplicates from Different Columns

You could implement a query using row_number() to delete everything but the most recent row. This partitions the data by the employee_id and orders it by the autoId column, then you delete everything that is greater than the first row number:

;with cte as
(
  select [EMPLOYEE_ID], [ATTENDANCE_DATE], [AUTOID],
    row_number() over(partition by [EMPLOYEE_ID], [ATTENDANCE_DATE] 
                      order by  [AUTOID] desc) rn
  from dbo.ATTENDANCE
)
delete 
from cte 
where rn > 1;

See SQL Fiddle with Demo

Why not just create the table column with a case-insensitive, accent-insensitive collation? This prevents duplicates according to the collation rules, and allows the sort of searches you seem to require:

CREATE TABLE Test
(
    col1 varchar(30) COLLATE SQL_Latin1_General_CP1_CI_AI PRIMARY KEY
);

-- Success
INSERT dbo.Test VALUES ('Gagné');

-- Failed, duplicate key
INSERT dbo.Test VALUES ('Gagne');

-- Success
SELECT * 
FROM dbo.Test 
WHERE col1 = 'gaGNE';

As an important side note, I should mention that T-SQL scalar and multi-statement functions have very poor performance characteristics. Using them in a computed column definition is even worse. Avoid.