You Question is incomplete. You have asked for populating the records from WebApi to jQuery DataTable, But writing cde for deleting a row fro the DataTable.
The following code may work for pupulating the data (assuming Col1 and Col2 are two cloumns for the DataTable)
$("#customers").DataTable({
serverSide: true,
processing: true,
columns : [
{ data : 'Col1' },
{ data : 'Col2' }
],
ajax: {
url: /api/customers/" + button.attr("data-customer-id"),
dataSrc : ''
}
});