Could you print the shape out logps[0]? It should be [batch_size, nb_classes].
I also just realized, that you are assigning your Sequential classifier module to model.classifier.
If you are using inception_v3, you should use model.fc instead.
Here is a minimal code snippet which should work:
model = models.inception_v3(pretrained=True, aux_logits=True)
criterion = nn.CrossEntropyLoss()

x = torch.randn(2, 3, 299, 299)
target = torch.empty(2, dtype=torch.long).random_(1000)

output = model(x)
loss = criterion(output[0], target)
loss.backward()

print(model.fc.weight.grad)

Could you try to compare your code to this snippet?