There is no doubt that this letter is by Peter, as shown by the same idiosyncratic terminology as that used by Peter in the Gospels and Acts, such as "tree" for the cross. The Greek is very good; Peter probably dictated to a skilled scribe, perhaps his assistant Silas. The introduction identifies the author and intended audience.
The epistle starts by discussing what a Christian is, then goes on to how one should behave, and finally focusses on relationships. There is much talk of suffering, but there is no evidence that the context involved particular persecution, so it seems possible that he has in mind the tension of being counter-cultural ("in the world but not of the world"). The Roman Empire had created a culture in which, as today, people were quite mobile, and many lived in a culture that was unfamiliar.
Peter at first rejected Jesus's offer to wash his feet (John 13:8), but learned the lesson that "unless I wash your feet, you have no part in me". Being a Christian involves letting Jesus serve us. Peter often returns to that lesson in this epistle.
Chapters 1–2: what a Christian is like;
Chapters 3–4: how Christians relate to the world;
Chapter 5: how Christians relate to each other.
© David Billin 2002–2019