Assessment doesn't matter at all. It's totally irrelevant.
What matters is what the house is worth. That means considering the comps, then subtracting for appliances and other major items. (Door knobs are cheap. So are outlet plates, etc., which a lot of people also seem to like to remove. Depending on what you want to spend for lighting fixtures, it could be anywhere from $25 to $100 per fixture. Not that much.)
So, forget asking price. Forget assessment. Look at the comps. Then subtract the needed repairs/replacements. It's true, as noted below, that banks generally aren't too flexible on an asking price. On the other hand, if it doesn't sell, they'll drop the price. I've heard plenty of stories of REOs--let's say priced at $300,000. Someone makes an offer of $290,000 and the bank rejects it. A month later, the bank drops the price to $285,000. So, if you really want the house, be prepared to go up to the lower of: (1) what the bank is asking, or (2) comps minus repair costs.
It really doesn't matter if you use that same bank for your own financing. In fact, it's much better to go in preapproved. And that way, if this purchase doesn't work out, at least you'll be prepared to make an offer on another home, whether foreclosure or otherwise.
Hope that helps.