Well, I dont know if you are from Seattle, but here is a rundown of some popular neighborhoods to look in.
Its all about how far you are willing to commute. For post-grad you may not want to live in the U-district (the surrounding neighborhood, most housing is north of campus). There are a lot of good/cheap eating options on the Ave, you are right near U-village for shopping if thats what you are in to, and there are plenty of bars, but besides U-village the area is kinda dumpy. Sure I loved my 4 years being in/near greek-row as an undergrad, but at this point in our lives I'm done with that lifestyle at least. This is also one of the heavier crime areas due to the low income housing and half-way houses conveniently scattered throughout the area. Lots of transients too.
Green lake is a good place to live, a lot of post-grads and young professionals live in that area. The lake itself provides a nice park for jogging, biking, roller blading (yes, people roller blade green lake!), and there are some decent eateries in the area. It's a straight shot on a bus to get you to right near the dental school, though you may need to hoof it a little bit through the main campus.
Ballard and Fremont are both awesome, but require a bit more of a commute, don't know the bus routes as well but UW is a pretty main hub for buses so I would hazard a guess that there is a bus that takes you to the general campus area. Ballard and Fremont are where most upperclassmen/post-grads/recent grads etc. tend to go out if you are looking for a nice balance to schoolwork with your social life. They are both pretty eclectic, and have their own farmers markets and street fairs.
Montlake is convenient for location, but you have to hoof it quite a ways or bus to get to any sort of entertainment/social atmosphere.
Capitol Hill is a cool area, but kinda pricey. To be honest I really didn't go up there often though.
Wallingford is another decent area to look for appartments, its between UW and Fremont in terms of location so its a little closer, but its not far to get to some fun spots to hang out and its close to gas works park, a summer time hot spot.
Eastlake can be nice, it puts you halfway between UW and downtown Seattle. Not a whole lot going on, but you can get some nice views of lake union. It's a really straight/easy shot to get to UW though by bus/bike.
If you have any questions feel free to ask, either by PM or on here. I definitely enjoyed my 4 years of undergrad, and it's unfortunate I won't be able to use my knowledge I've gained since I'll be going to Tufts, but I don't mind sharing it with those that ask! IF I had gotten in to UW, it would have been between Fremont, Green lake, and Ballard for me. Fremont and Ballard are cooler, but Green lake would maybe win due to convenience.