The "clustered index" is the actual table. All of the columns of the table are in the "clustered index" (with some exceptions for "out of row" storage for lobs, etc.)
When you change the value of a column, it has to be changed in the table pages, as well as in any index that the column appears in.
In terms of performance for quickly locating the rows to be updated (for your particular query), an index on dbo.Table(Column1,Column2)
or dbo.Table(Column2,Column1)
would be the most appropriate.
If it's possible that the column being modified already has the value being assigned (i.e. @Param1 and @Value both represent the same value, then adding another predicate may improve performance by avoiding a lock being obtained on the row.
UPDATE [dbo].[Table]
SET Column1 = @Value1
WHERE Column1 = @Param1
AND Column2 = @Param2
AND Column1 <> @Value1