Islington is a great choice. I live up near Highbury, right on the park, and it's probably one of the most pleasant places I've been in London, especially with the proximity to Upper Street. If you like being near a bunch of great restaurants, pubs, boutiques, and so forth, Upper Street can't be beat. Some neighbourhood names that you may see for the areas around here are Highbury, Canonbury, Angel, Clerkenwell, Barnsbury, and so forth.
Sadly, £400/week is probably about the average for flats around here. The overall inventory is limited and this is one of the most desirable boroughs in London; you won't necessarily end up with a luxurious flat for that price, but it should be entirely comfortable, with enough searching. The prices seem to get better the further from Upper Street you get, though. Having a dog, however, will be a *huge* negative. I've wanted to get one for awhile, but the estate agents have universally agreed that it makes flat hunting basically impossible, especially in areas where the owners/landlords can be picky (like Islington). Keep this in mind, when you're searching, as you may need to make significant compromises to find a place.
Regarding how to look, it's a bit of a nightmare hunting for flats here. Whereas Craigslist is pretty ubiquitous in the US, at least in the major cities, there doesn't appear to be anything here with even remotely similar levels of penetration. The few aggregation sites that exist (Globrix, Zoopla, Findaproperty, Gumtree) are mostly fed by estate agents and each has only a partial view of availability. Basically, you have to browse each of these sites, as well as the websites of pretty much every estate agent you know, to see what's actually available. Around Angel, it's not uncommon for a flat to be let before the estate agent has even bothered posting it online, so it really helps to pop into each of their offices and let them know what you're looking for; they may tell you about properties before they're uploaded.
As a point of reference, it took us about six weeks to find and settle on our current flat, and we already live in London. There isn't much available, a lot of what is available is garbage (our range was £350-£450), and it's a very cut-throat segment. We lost out on two offers because people were willing to pay MORE than the already-inflated asking price (a lot of City bankers live around here, given its proximity, and they seem to have no financial worries at all). With that said, it's still useful to negotiate; our current flat was listed for £475 and we ended up getting it for £425.
Although a lot of this sounds negative, once you actually do find a place, you'll fall in love with the area. It's truly a wonderful place to live.
P.S. As was already said, avoid Foxton's... they mark their properties up by at least £75 and can be pretty evil. If you're in love with one of their properties, try looking for it elsewhere; we've seen probably 20-30% of their inventory listed elsewhere, and often for far less.