Broken Iphone 5s? It only shows life if i plug it into my laptop. Low battery symbol followed by apple logo then dark blue screen. Fixable?

When I plug my Iphone 5s into my laptop, it shows the low battery symbol for about 4 seconds, then the white back screen with the black apple symbol for a brief second then goes to dark blue screen for the rest of the time…

what's going on with this phone? Is there any hope I can get it fixed?
I have so many pictures I would like to save.

Assuming it's never been damaged / drowned, either the battery needs replacing or it's just too low for the phone to start up.

Leave it connected for at least a couple of hours then press home and power at the same time for about ten seconds, until the Apple logo re-appears. That usually restarts them if they have crashed due to a dead battery (and does not erase anything).

If that does not work, a new battery kit is very cheap - eg:
http://www.ebay.co.uk/...1971211070

I would take it to an Apple Store or independent apple repairer. If it is anything more than a battery the Apple Store will suggest a refurbished phone and no data recovery. An independent repairer should at least attempt to get your data back and repair it
Just google iPhone 5repairs and your city and you should get some hits