It almost seems presumptuous to try to give details here, since this is one of the most famous climbs in the world and probably much better documented elsewhere. In any case, the Supertopos description and topo give all you could want in terms of pitch-by-pitch beta, so just a few notes that might help clarify things:

• There has been some rockfall in recent years that impacts the Death Slabs approach, so you might want to ask around before heading up, but if they are in decent shape and you're comfortable with fourth-class scrambling, this is definitely the way to go, but don't even think about trying this in the dark if you have done some recon in advance.
• Doing Half Dome in a day is a rather big enduro day, but the climb itself isn't that long. If you have done the DNB on Middle Cathedral in a day and the South Face of Washington Coumn in a day, then you can probably do Half Dome in a day (count on about as much time as the DNB and 1.5 to 2 times as much as the South FaceThe part that kills you on Half Dome is the approach and descent which are both much longer than the DNB, so if you don't have those dialed in the dark or don't have that good eighteen-hour ultra-marathon endurance, plan to bivy at the base on one or both nights.
• Stuff comes off Half Dome, so if you do bivy at the base, pick your bivy spot carefully. It will be worth it to hike up a short ways and have a few minute approach in the morning.
• Go light, especially if you're doing it in a day. The .5 and .75 Camalot size is much in demand and having three each is handy, but in other sizes you will probably just want two each and then one #3 and one #4 (this assumes that you are a comfortable 5.10 free climber, weaker climbers may want a couple more large pieces). Lightweight aiders are ideal and you'll often use just one. Practice transitioning from free to aid and back again, because that skill is essential to moving well and having fun on this route. If you're a 5.10 climber, you'll spend little time in the aiders as you can free or French free much of the route. I believe I only pulled out both aiders for a couple of short sections and Theresa aused two aiders on the Robbins traverse.
• There is a ton of fixed gear on this climb and just because it's there doesn't mean you have to clip it all and this isn't A5. Pack a few more biners, slings and draws than you would for a free climb, but not heaps more.
• There is usually water in the spring at the base until late summer, but it is not the most inspiring place to draw water from. I usually do not filter my water in the Sierra backcountry, but I don't like to drink from here without a filter.
• Afternoon thunderstorms are common on Half Dome. Though usually they are small and the pass quickly, you could easily get caught in a pretty good storm and find it pretty cold afterwards. Remember, you will be almost 5,000 feet above the Valley and it can be cold any time of the year in a storm. This is not a climb to forget your raincoat unless you know you will be fast and off before the storms hit (do you really know?). Of course, lightning strikes are a real danger too, but your clothes won't help you there.
• There are ample opportunites to passing on this route, so you don't need to worry too too much about being the first ones on.