I researched this a while back and even went so far as trying it out. Your blood is going to coagulate before you bleed out. You need something to suck out the blood and you need the blood in the needle not to contact air (which you'll notice is what they do when you give blood).
Here's how I thought of doing it.
Get a bag of medical saline solution with a valve in the top and one on the bottom. Make sure the bottom valve's outflow is about what the outflow from your blood draining needle is.
Tie off your arm with a piece of surgical tubing and pump your fist, like you do giving blood.
Have the needle attached to the drip line from your saline bag.
Lie down, with saline bag LOWER than you.
Open outflow valve a little bit.
Inject the needle into your elbow vein. This will probably take a lot of practice before you can do it.
Open drip valve so blood is now flowing from your vein into the saline bag.
Control of outflow is key here, or you risk creating too much suction and collapsing the vein.
I would suggest donating a lot of blood, first, and paying close attention to everything they do until you can do it yourself.
I also think you should take anticoagulants before doing it (aspirin is one - I am sure there are better).
Also, the day you will do it, go donate blood as many times as possible. In my city, they don't ask too many questions. In fact, you might try to donate blood at different clinics in the days running up to your attempt. In other words, have the professionals get a liter or two out of you first.
I suggest practicing letting blood out with needles and hitting your vein in the shower. It is cathartic to splatter the walls with blood and looks really dramatic. Always wipe down the injection site with alcohol and use clean, disposable needles. Don't let any loved ones catch you doing this as it will really freak them out.
This seems to me to be a peaceful, if slow way to go, but lots can go wrong. However, not many side effects if it goes wrong.
