An Akathist has a certain basic structure. I would not take the Akathist or Canon to the Theotokos as the prime example since that is somewhat more complex and rich.
First, the Kontakion is written, which begins the akathist and usually sums up the entire "theme" of the Akathist. This is usually repeated throughout the akathist- or sometimes just once at the beginning, once at the end.
Then, you compose the verses for the feast/saint/theme.
Then there is a short repetitive refrain like "Most Holy Theotokos Save us" said/chanted between each stanza or verse.
From what I observe, every 5 to 7 verses, Alleluia (3x) is inserted at the end of the verse. Someone with more liturgical info can elaborate on this.
Then, at the end, there can be a prayer to Christ, Panagia, the saint to who it is dedicated to, etc.
Thats a very basic explanation of it. It's not really that hard if you already have something in mind or are good at writing.