two ways

Option Explicit
Sub sum_column_H()
Dim lastrow As Long
Dim ws As Worksheet

Set ws = Worksheets("sheet1")
lastrow = Cells(Rows.Count, "H").End(xlUp).Row
ws.Range("H" & lastrow).Offset(1).Formula = "=sum(h5:h" & lastrow & ")"
End Sub

or if you want just the value use the line below instead
ws.Range("H" & lastrow).Offset(1).Value = Application.Sum(Range("h5:H" & _
    lastrow))

Assuming there is nothing below your data in Column H, maybe:

Sub test2()
With Worksheets("Sheet1")
 If IsEmpty(.Cells(.Rows.Count, 8)) Then
  With .Cells(.Rows.Count, 8).End(xlUp)
   .Offset(1, 0).Formula = "=Sum($H$5:" & _
    .Address & ")"
  End With
 End If
End With
End Sub

Assuming there are no breaks in your data and H5 could vary, you might also
try substituting this into the above:

.Offset(1, 0).Formula = "=Sum(" & _
   .End(xlUp).Address & ":" & .Address & ")"