What does it take to get through a PhD's worth of reading? Surgical ruthlessness. Become relentlessly efficient with your reading and the sheer vastness of the literature will begin to seem manageable. I do spend a little bit of time each day getting 'on top' of my reading: I have to stand on the massive stack of papers in my bedroom to reach my shoes on top of the wardrobe. This is a very dangerous use of the literature. A good PhD student knows that any pile of photocopied papers is inherently unstable and liable to rearrange itself into a random order at the slightest provocation.
The first rule of ruthless reading is to be remorseless about organizing regular hunting trips into the literature. Citation and journal databases like the Web of Science  and referencing software like Endnote  make literature searches and reference-handling a piece of cake. But, however good your search software, you still need to know exactly what you are searching for. Compile a definitive list of your key words and try including important authors' names as well as key scientific terms. Then it's just a matter of remembering to run your search string to pull out all the new articles. These periodic blitzes are a good way to keep up-to-date with the mass of the literature. As an alternative to blitzing the database, some journals offer free e-mail alerts to help keep you posted. This might be worth doing for a few major publications, but subscribe to too many minor journals and your inbox will soon be bombarded with endless pages of contents that you'll probably never bother to look at.
Maybe it's the nagging doubt that your search string is too restrictive, but don't ignore the occasional compulsion to enter the library and just let your instincts guide you. It's amazing how often I've stumbled on something directly relevant, apparently by good fortune. You may scoff, but it works!
Another great way to get yourself quickly in the picture is to start your search in the references section of your top 10 favourite papers. Chasing relevant citations can lead to one almighty paper chase, but I've turned up some really spot-on stuff in this way. If you can find a lab that maintains an up-to-date list of references for your subject on its Web page (i.e., not just publications from that lab) so much the better.
If the first rule of ruthless reading is search hard, the second is to actually get around to reading it all. Of course, we should all be reading the really key papers as soon as we find them, but what about the other, less obvious, literature--the background reading? With so much to do in the lab, how can you possibly keep up with the insidiously growing list of stuff to read? And how do you begin to select what to include on your final reading list and what to ignore, store, or delete? If you are anything like me, you photocopy or download papers first and ask questions later. This has two disadvantages. First, you'll amass a vast, not to mention slippery, pile of paper on your bedroom floor. Second, you'll soon not have a clue what gems and mysteries lie within the pile, as yet unread. To make matters worse, when you eventually pluck up the courage to start sifting through it, and believe me that can take some time, you face another problem. You are guaranteed to freak out at how long it will take you to read the stuff that catches your eye just in the first two inches of the stack. If you have one of these piles, it must be counted as one of the worst possible PhD monsters. You need to slay it before the mere thought of it begins to spook you.
Tackling your monster pile is straightforward, if difficult. Catch yourself in a decisive mood and dive in, sorting into piles labelled 'Essential', 'Nice to read if I get time', and 'Why did I even photocopy/print this?' It helps to put the papers in order so you can easily find the one you need. Use first author surnames, as you'll doubtless have your references in this order.
Even your ruthlessly reduced reading list may be quite a stack. If you are a slow reader, you might want to try to increase your reading speed . In a nutshell, you have to aim to read half a line, or even a whole line in one glance, and keep moving. Avoid rereading the same phrase or passage--you may do this more often than you realize. As your reading speed increases, you'll almost certainly increase your level of comprehension. This is because whole sentences make more sense to your brain than a string of individual words. With practice you'll find yourself hunting the text for key words or phrases which ring a mental alarm to slow down and read more carefully. In all but the most earth-shattering papers (for this sort I still have to sit down and read slowly) you'll probably be looking for no more than half a dozen pieces of information. Another good tip if you are pushed for time is to read the first and last sentences of each paragraph until you track down the section most relevant to your needs.
Find your own best time and place to read. Naturally, forget it if you're too tired or it won't sink in. I find the best policy is to read an important new paper quick and hard quite literally as soon as I get hold of it. Then I've at least got a grasp on it for a later in-depth read. For less pressing stuff I often read through a paper in the office or lab (whichever is quieter) during a short break, say 20 minutes, between experiments. You'll be amazed how many accumulated papers can trickle into your memory in the space of a few weeks.
Finally, don't neglect the value of a good review article. You might have to take the views expressed with a pinch of salt, but they're a great way of getting a quick grip on whole chunks of literature. After all, a little reading goes a long, long way!