I shoot with a NEX-5N, and a NEX3 before that. I also use the 55-210mm lens, though I also stack a 1.7x teleextender on the end of it to get more reach as with bird photography, you can never have too much zoom!
The very basics would be: make sure you are using AF-S focus mode, and switch to Flex Spot Focus mode (if you weren't already) set to the central focus area. This is the very smallest focus box available, and will help you focus specifically on a smaller target in front of a busy background. I should mention that you shouldn't be in 'auto' mode on your main shooting mode selector - auto mode doesn't let you make changes to various settings...by switching at least to 'P' mode (program auto), or the A, S, or M modes, you can then take control over your various focus modes and settings.
Another tip - strange as it sounds...turn OFF the focus assist beam light. It won't help you and often hurts you by slowing down autofocus, and sometimes causing autofocus to miss the intended subject. When focusing on smaller birds against a busy background, be sure to try to look for something like the bird's eye, or a contrasty pattern on the bird's head, to try to focus on. Place the focus box over that spot so that the line of dark/light contrast passes diagonally or vertically through the focus box, which will allow the camera to find the contrast and focus on it. Always use a half-press technique with the shutter button to get an attained focus confirmation beep before pressing the shutter the rest of the way to take the shot - this little pause will let the focus system do its job and settle, and also allow stabilization in the lens to work itself out...if you simply mash the shutter button it forces the camera to try to quickly focus AND shoot all at once, and this can cause it to give a false focus reading and clearing the camera to shoot.
Hope this helps. You'll get better over time - bird photography can be very difficult, and learning to focus with and control any camera will always be a bit of a learning curve. Even full-frame pro DSLRs can't 'guess' what you want to shoot and can focus on the wrong thing unless you learn to feed the right focus area to the camera. Eventually, you should be able to thread the focus through small gaps in branches, in front of busy backgrounds, even focus on moving birds in flight...it just takes practice and the right settings!
