I wish I could be of more help, especially because I could not agree more that learning Hebrew is integral to fully experiencing Judaism, and to comprehending Jewish text. But the truth is, Hebrew is a hard language to try and teach oneself, especially if one does not already know a non-Germanic/Romance language.
I'm not saying don't try anything on your own. If you want to do a Pimsleur course or a Rosetta Stone course in modern Hebrew, it would be a good start, no question. But Hebrew is a tough nut to crack alone. And, though this may be secondary, keep in mind, modern Hebrew isn't quite the same as Biblical Hebrew, especially given the Bible is written so extensively in poetry. It would be manageable to transition from one to the other, I think, if one had had very good immersion-quality training in modern Hebrew, but with what most of us could manage as autodidacts...? Chancey, I'd say. Not impossible, but hard.
It may be inconvenient, unhelpful, or expensive advice, but I really recommend taking classes. An
ulpan (immersion course in Hebrew) would be ideal. Whatever you do, I would take Hebrew from Jews. If you are looking at Hebrew classes at a non-Jewish institution, make sure they have a Jew teaching the course.
I know, call me insensitive, call me rigid and exclusionary, call me judgmental, but in my experience non-Jews rarely seem to be able to perfectly master the rhythms, feel, and mindset of Hebrew enough to be language teachers-- teachers of Jewish text, theology, poetry, etc., sure...but pure language...? It does happen, don't get me wrong, but I wouldn't want to gamble on it the first time out.
Maybe if you were taking the class someplace absolutely unimpeachable, like SOAS (who
do offer a course, if you're interested).... Otherwise, I do notice that
LJCC offer a series of courses that look promising, and
LSJS (of which, I admit, I have heard only little) also do so.
Beyond this, I would ask at the
New North London Synagogue for where they recommend taking Hebrew. They will surely have good advice for you.