You are almost there, however you need to exclude the cell end character from the cell range text or you will never have a text value of "0". Also having named the bookmark as a range, then use the name later in the macro to delete the range.
Dim oBMName As Range 'Use a name that doesn't conflict with your bookmark name
Dim oCell As Range
'Define the bookmark range
Set oBMName = ActiveDocument.Bookmarks("bmName").Range
'Define the cell range to evaluate
Set oCell = ActiveDocument.Tables(1).Cell(1, 2).Range
'Remove the cell end character from the range
oCell.End = oCell.End - 1
'Check the range text content
If oCell.Text = "0" Then
'Set the range text to nothing
oBMName.Text = ""
'Reassign the bookmark to the empty range (in case you need it later)
ActiveDocument.Bookmarks.Add "bmName", oBMName
End If