At, by, in the end of this week

The third is ungrammatical and should be discarded.

The first two are just about grammatical but mean different things.

By the end of the week means you will have completed it by then.

At the end of the week means you intend to start reading it then.

The problem with the grammaticality is the use of the present progressive, when it seems you need the future. Ideally you need to say:

I will read it by/at the end of this week.

At the end of the week means you gonna read it at the weekend whereas by the end of the week means you're currently reading and will possibly read it till the end ( also consider that BY is generally used with Past and Future Perfect tenses). I haven't got any idea about in the end of the week. If you ask the general difference, in the end equals to finally, and at the end denotes the meaning known to everybody.