Learn Russian: Как добиться независимости в бизнесе?

The verb добиться (to achieve / to attain) strictly requires its object to be in the Genitive case. This is known as the Genitive of Achievement. Verbs expressing the idea of reaching a goal, desiring, or striving for something usually take the Genitive rather than a direct Accusative object.

Why does the verb добиться have the reflexive -ся particle? Does it mean to achieve oneself?

No, the -ся here changes the meaning of the base verb rather than acting as a true reflexive. The base verb добить means to finish someone or something off. Adding -ся shifts the meaning to reaching a goal through your own intense effort. Because it has -ся, it cannot take a direct Accusative object, which is another reason it forces the Genitive.

I often mix up noun endings. Why does независимость take an -и ending in the Genitive?

независимость is a feminine noun ending in a soft sign (-ь). This puts it in the 3rd declension group. For all 3rd declension nouns, the Genitive, Dative, and Prepositional singular endings are always -и.

There is no subject in this sentence. Why use the infinitive добиться directly after Как?

Using a question word followed immediately by an infinitive is a standard Russian structure for general, impersonal questions. It translates perfectly to the English 'how to achieve'. You do not need to add a pronoun like 'you' or 'one' when asking for general advice.