Learn Icelandic: Hún klæðir sig alltaf mjög vel.

In English, we just say "she dresses," but in Icelandic, the verb klæða means "to dress someone." If you are dressing yourself, you have to use a reflexive pronoun. So, hún klæðir sig literally means "she dresses herself."

Why do we use sig instead of sér here?

The verb klæða requires the accusative case for the person being dressed. Sig is the accusative form of the third-person reflexive pronoun, while sér is the dative form.

Why does alltaf come right after sig?

In Icelandic, adverbs like alltaf (always) usually come after the main verb and its direct object. Since sig acts as the object here, alltaf follows it.

Could I use the word for "good" instead of vel here?

No, just like in English, you must use the adverb vel (well) to describe how an action is done. You would only use the adjective for "good" to describe a noun, like saying a person or a piece of clothing is good.