Reddit mentions of Biblical Hebrew for Students of Modern Israeli Hebrew

Here are some book recommendations:

• Biblical Hebrew for Students of Modern Israeli Hebrew, Marc Zvi Brettler
  
• Hebrew Verb Tables, Asher Tarmon and Ezri Uval
  
• For those who already understand Hebrew well enough: מילון אבן שושן המרוכז (great dictionary, and there is a grammar reference in the back)

It's certainly possible, but you would have to keep in mind that they are different. I'd recommend starting with Modern Hebrew. (I'm not sure exactly which book to use, but Ha-Yesod looks decent.) After you've done the equivalent of two college semesters of Modern Hebrew, use Biblical Hebrew for Students of Modern Israeli Hebrew by Marc Brettler to learn Biblical Hebrew.

The reason I recommend starting with Modern Hebrew is because Marc Brettler's rocking awesome textbook assumes you've stared with Modern Hebrew. Additionally, you're probably more likely to find resources about Modern Hebrew aimed at a new learner, much like how anyone might start to learn a new language. In contrast, in my small experience (seriously, it's small, because I learned in Hebrew school instead of online or through a book), beginner's Biblical Hebrew will likely be taught as if it's some Latin-like dead language without the usual practices and with using confusing, archaic descriptions.

As was mentioned, Ashkenazi Hebrew / Sephardic Hebrew are really just pronunciation differences, there's no real syntactic or grammatical differences.

To put simply (and avoiding the question of what exactly Biblical Hebrew is as it is not necessarily a unified language but a hobgoblin of various significantly diverse dialects and Aramaic) , Biblical Hebrew is to Modern Hebrew as Shakespearean English is to Modern English. Learning Modern Hebrew will make it exponentially easier to learn Biblical Hebrew. And, once you learn it I would suggest this book: http://www.amazon.com/dp/0300084404/?tag=stackoverfl08-20