Your possibilities are only limited by your time and money. I've gone twice, and to two very different types of places. Anywhere you go you will definitely have the time of your life. Both times I went with a group of guys, which makes it much more enjoyable, and usually cheaper. The first trip was to Soldotna and the Kenai River. We flew to Anchorage rented an SUV and drove to Soldotna. We stayed at the Eagle's nest B&B which wasn't on the river, but was very nice. Our guide service was Wes the Samurai. He made all the fishing arrangements, as we did 3 days of Salmon fishing and one day of halibut fishing out of Homer. You can figure on about $300-$400 a day for guide service and accomodations, so a week should run you about $1500 - $1800, plus airfare and car rental.
The other trip was very different. This had to be the ultimate fly fishing trip. We flew into Anchorage and took a small plane to Anvik. After about a 2 hour jet boat ride up the Anvik river we arrived at the Anvik River Lodge. This place to say the least is pretty close to heaven. All meals are included, and were pretty much gourmet. The last night we had crab legs as big as my arms. The fishing was phenomenal, and you have the whole river to yourself. We caught arctic char, greyling, chum salmon, and kings all on flies. This trip will run you about double for a week, but if you ever have the chance I would highly recommend. Good Luck