What's wrong with "within this week?"

While it sounds a bit stilted, there is nothing wrong with it.

A more common expression (in US usage) would be

I want to finish it within the week.

Said in that fashion, it means this week. Exactly what this week means may be open to interpretation. It might mean the work week (usually Monday to Friday), the calendar week (usually ending on Sunday, but for some on Friday, and for others Saturday).

Both the original sentence and the offered alternative would indicate that the job should be done by the last day of the week we are currently in, regardless of the day the instruction is given. It would be unusual to use the expression unless there were more than one day left in the week.

However, if you said

I want it finished within a week.

that would likely be interpreted to mean within seven days (or possibly five work days) regardless of the actual day of starting and stopping.

correct in reference to ... grammar, style, or culture?

Grammatically, it's fine. I want to say its antiquated grammatically but I don't have a reference for that.

Stylistically and culturally, it's odd and shouldn't be used except in jest. It has a certain theatrical flair to it since it is unnecessarily verbose.

As suggested in the other answer:

I want to finish it this week

is probably the most natural.

or

I want to finish it before the end of the week

but the second formulation will introduce an ambiguity as to whether one includes the Saturday.