Intact iPhone's display broken, how to make backup from locked phone?
It depends on if the both the digitizer and the display (LCD) are broken or just the display. If it's just the display, then you should be able to interact with the iPhone, even though you won't be able to see it. In other words, treat the iPhone as if it were fully functional and swipe where Slide To Unlock should be and then type in your passcode by tapping where the buttons should be.

If it is both, it will need to be repaired. If it is an original iPhone, an iPhone 4, 4S, 5, 5C, 5S, 6, or 6 Plus you will need to buy a digitizer assembly , not just the "glass" or the digitizer or LCD separately. If it is an iPhone 3G or 3GS, you can buy the parts separately. As far as parts go, don't buy a $5 digitizer assembly someone made in their garage in China, the best thing is to pay $25 or so and get a good one (not to say that bad parts don't exist at that price).