django rest framework order listview by serializer specific field

You could try overriding the get_queryset method on your View and add an annotated field on your queryset for count_need. I think this would be preferable as it may be faster in SQL for large lists, and works with the built-in OrderingFilter of rest framework.

Example:

from django.db.models import Count

class MyView(APIView):
    def get_queryset(self):
        qs = self.queryset
        qs = qs.annotate(count_need=Count('need'))
        return qs

Recommended Reading:
https://docs.djangoproject.com/en/1.11/topics/db/aggregation/#aggregations-and-other-queryset-clauses

The other option is to implement some custom ordering in Python instead of using the Django ORM. You could do this by overriding the list method on your ListAPI view.

def list(self, request, *args, **kwargs):
    queryset = self.filter_queryset(self.get_queryset())

    page = self.paginate_queryset(queryset)
    if page is not None:
        serializer = self.get_serializer(page, many=True)
        serializer_data = serializer.data
        sorted_serializer_data = sorted(serializer_data, key=lambda x: x['count_need'])
        return self.get_paginated_response(sorted_serializer_data)

    serializer = self.get_serializer(queryset, many=True)
    serializer_data = serializer.data
    sorted_serializer_data = sorted(serializer_data, key=lambda x: x['count_need'])
    return Response(sorted_serializer_data)