I'm not sure if there is a way to do it while your typing. I think excel generally stretches the cell view to display all the text before it fires the worksheet_change event.
This code will resize the column after you have changed and moved the target to a new range. Place it in the worksheet module.
Private Sub Worksheet_Change(ByVal Target As Range)
Dim nextTarget As Range
Set nextTarget = Range(Selection.Address) 'store the next range the user selects
Target.Columns.Select 'autofit requires columns to be selected
Target.Columns.AutoFit
nextTarget.Select
End Sub
If your just looking to do it for a particular column you would need to check the target column like this:
Private Sub Worksheet_Change(ByVal Target As Range)
Dim nextTarget As Range
Set nextTarget = Range(Selection.Address) 'store the next range the user selects
If Target.Column = 1 Then
Target.Columns.Select 'autofit requires columns to be selected
Target.Columns.AutoFit
nextTarget.Select
End If
End Sub