Learn Arabic (Modern Standard): أنا أصدقك.

It is attached to the end of the verb! In Arabic, object pronouns (like "me", "you", or "him") attach directly to the end of the verb. Here, the letter ك at the very end of أصدقك is the attached pronoun meaning "you".

Does this sentence mean "I believe you" to a man or a woman?

It can be either! Because Arabic is usually written without short vowels, the final ك could be pronounced as -ka (when talking to a man) or -ki (when talking to a woman). The written text is exactly the same for both.

If the verb already starts with an alif for "I", why do we need أنا at the beginning?

You do not strictly need it. The verb أصدق already means "I believe" all by itself. Adding the separate pronoun أنا (I) just gives it a little extra emphasis, like saying "I am the one who believes you." You could simply say أصدقك and it would mean the exact same thing.

How do I pronounce the middle part of أصدقك?

The verb is pronounced "usaddiqu". There is a doubled consonant in the middle (called a shadda on the letter daal), so you hold the "d" sound slightly longer: u-sad-di-qu. Then you simply add your attached pronoun (-ka or -ki) to the end to make it "usaddiquka" or "usaddiquki".