When you ask for a favor you should be direct, succinct, charming, and—whenever possible—entertaining.
Obviously this is easier said than done and you won’t always get it right. You also can’t win over everybody, so don’t expect that you will.
Next time you need to ask someone for a favor, just remember to set aside some time beforehand and think of a way you can make them smile. If you can do that and get right to the point, you’ll be in good shape.
The example below is a good way to ask for help:
I’m hoping you could vote for me in an online contest where I can win a big prize for my work. I haven’t seen you in years. I hope you’re doing well. Congratulations on all you’ve been doing. It’s really amazing! We should grab coffee sometime.