You need to select a point on the map first, it will appear at bottom of the pop-up in the map when selected.
Otherwise, if you still do not see a delete button, it could be that you do not have permission to edit (more importantly delete features) from the feature service?
What are the capabilities of your feature service, if you open the service url of the layer and then view the JSON link at top of page, scroll to the bottom you will see the capabilities, it should look as follows
"capabilities" : "Create,Delete,Query,Update,Editing,Extract"
If you do not have Delete listed it may mean you do not have permission to delete features in feature service.
Phil.