Learn German: Ich kann mir den Rücken nicht waschen.

When you do an action to yourself, you usually use the accusative reflexive pronoun (like mich). But if there is another direct object in the sentence—in this case, the back (den Rücken)—that object takes the accusative case. The reflexive pronoun then gets bumped down to the dative case (mir) to show for whom or to whom the action is being done.

The English translation says "my back", so why doesn't the German sentence use meinen Rücken?

In English, we use possessive words (my, your, his) for body parts. In German, it is much more natural to use a dative reflexive pronoun along with the definite article (der, die, das). Saying Ich wasche mir den Rücken literally means "I wash the back for myself". Using meinen Rücken here would sound a bit unnatural to a native speaker.

Why is it den Rücken and not der Rücken?

The word Rücken is a masculine noun (der Rücken). Because it is the direct object of the sentence (the thing being washed), it has to be in the accusative case. For masculine nouns, der changes to den in the accusative case.

Why is the main verb waschen at the very end of the sentence?