Quote OP:
<input data-validators="required" ...
"As you can see, I use a custom attribute named data-validators. How
  can I use the jQuery Validation plugin for this case?"
You can't.  The plugin was not written to recognize that attribute.
See:  http://jsfiddle.net/y5xUF/
However, you can define validation rules as per the following methods:
1)  Declared within .validate()
$(document).ready(function() {
    $('#myform').validate({
        rules: {
            fieldName: {
                required: true
            }
        }
    });
});
http://jsfiddle.net/uTt2X/2/
NOTE:   If your field name contains special characters such as brackets or dots, you must enclose the name in quotes...
$(document).ready(function() {
    $('#myform').validate({
        rules: {
            "field.Name[234]": {
                required: true
            }
        }
    });
});
2)  Declared by class:
<input name="fieldName" class="required" />
http://jsfiddle.net/uTt2X/1/
3)  Declared by HTML5 validation attributes:
<input name="fieldName" required="required" />
http://jsfiddle.net/uTt2X/
4)  Declared using the .rules() method:
$('input[name="fieldName"]').rules('add', {
    required: true
});
http://jsfiddle.net/uTt2X/3/
5)  By assigning one or more rules to your own class using the .addClassRules() method:
$.validator.addClassRules("myClass", {
    required: true 
});
Then apply to your HTML:
<input name="fieldName" class="myClass" />
http://jsfiddle.net/uTt2X/4/
6)  Declared by field type:
<input name="fieldName" type="email" />
http://jsfiddle.net/gv0o6syt/