I'm fortunate enough to have a job that offers me the opportunity to undertake many adventures that most people never get the chance to try. Although it is far from my number one responsibility, I'm required to maintain a certain capability in both rock climbing and alpine mountaineering. As such, my coworkers and I completed a 4-day training event in Mount Rainier National Park that culminated with a summit. While a person with a solid background in safe mountaineering practices can reasonably set a goal of ascent -> summit -> decent from Paradise and back within a 24 hour period, spreading the climb over a several day period is far safer and is less physically taxing.
If you make the climb with Rainier Mountaineering Incorporated, I believe that they do most of their summits in three legs: Paradise (~5,200 feet) to Camp Muir (~10,000 feet), Camp Muir to the summit (14,411 feet) and back using the Disappointment Cleaver route, and then Muir back to Paradise. I don't know if they do the second two legs in one day or two.
A couple of things to keep in mind should you summit independent of a guide company... First, if you don't go with a guide outfit, ensure that you have several people in your group who are intimately familiar with sound alpine mountaineering tasks. There are a lot of things that can easily kill you up there if you don't know what you're doing. Second, ensure that whichever group you end up with (a guide or knowledgeable friends), they spend time teaching you the basics of glacier travel, rope teams, self arrest, crevasse rescue, avalanche procedures, etc. Finally, you are better off moving beyond Muir and making camp at Ingram Flats (~11,200 feet). You're more likely to beat the rush of the guided groups and not get caught in traffic jams if people ahead of you are climbing slowly.
The two months with the largest number of successful summit attempts are July and August. If you go during that period, however, you must be aware of the dangers that go along with it. Higher temperatures during the day lead to snow and ice melting. For this reason, your best bet is to begin your summit attempt at around 11 PM or midnight. Yes, you're doing most of the ascent from camp in the dark with headlamps, but it is still safer than the alternative. If you summit too late in the morning, there is a much higher possibility of having the ice bridge you may have crossed on the way up without even realizing it being melted out or much less stable on your way down.
When my group summited in August 06, we left from Ingram Flats and were the first to the top that day. The drawback is that the RMI guides who were probably bringing a fresh log book to the summit that day were well behind us so there was no book up top (the case was empty).
Aside from having some knowledgeable people in your group, you MUST have decent gear. At 7:30 AM on 25 AUG 06 it was brutally cold and windy up top. I would estimate that it was about 15 degrees fahrenheit and the winds were arong 50 mph. That kind of cold and wind combined with the disorientation of Acute Mountain Syndrome leads to the kind of bad decision making that gets people killed.
The Dissapointment Cleaver route is not technical at all. You can literally walk to the top and back if you know the route and practice appropriate safety.
Hope that helps some.