I wrote in a previous post how relative times work for days. (Similar to how we say tomorrow, yesterday, day before yesterday, day after tomorrow, etc. in English).
Well the options for weeks are a bit different to the ones for days. To start with, it's worth noting that there are two common words for weeks:
星期 (Xīngqí) is literally "star period" but means week. It's pronounced like "shing chee".
周 (Zhōu) also means week but can mean many other things as well.
I commonly hear 周 (Zhōu) used in 周末 (Zhōumò) which is pronounced somewhat like "Joe Moore" would be in English.
So let's look at the relative references:
这个星期 (Zhège xīngqí) is pronounced like "jer ger shing chee" and means "this week". Similarly you could use 这个周 (Zhège zhōu), and you might even leave out the middle character 个 (ge). I've also seen 本星期 (Běn xīngqí) used for "this week".
Last week is 上个星期 (Shàng gè xīngqí) is pronounced like "shung ger shing chee". Curiously the first character means "up" not "last".
And based on that, it's not surprising that 下个星期 (Xià gè xīngqí) which is pronounced like "shaa ger shing chee" means "next week". The first character means "down" not "next".
Often you'll just see people say 下周 (Xià zhōu) which is shorter and to the point.
Chinese is wonderfully concise at times, and 上上个星期 (Shàng shàng gè xīngqí) is a good example. It's the week before last. You can also just write that as 上上周 (Shàng shàng zhōu).
And no surprise, for the week after next, we can just say 下下个星期 (Xià xià gè xīngqí), or more commonly, the beautifully short 下下周 (Xià xià zhōu).