I have written a program with Qt5.3.1 and run it on my development machine where it works fine. I copied all necessary .dll files into the folder where my .exe is. These are:<
The short answer is you either need to do a static build, or you need to redistribute the Qt shared library DLLs. If you're looking for a smaller file size static builds are often better, but there may be legal implications with this approach if you're using Qt5 under a LGPL license. I'm not a lawyer, but you can read about LGPL linking rules in LGPL section 4.1.
The long answer is you should read the general Qt5 deployment guide and the Qt5 for Windows deployment guide. These go into quite a bit of detail.