How to properly use an op-amp?
Your 2nd datasheet snip is in mV not volts, and the output range is relative to the supply voltages. So with a 4.8V supply and 5K load (to 0V) the linear output range is 0.1 to 4.7V. If you bias the input and output at 2.4V you can get 4.6Vp-p. The op-amp output cannot exceed (or even meet) the supply voltages. 

If the input is biased at 2.4V, your input range is -0.3 to 3.6V so you can only handle an input voltage of 2.4Vp-p = (3.6-2.4V)*2, based on the input range, however you also need to make sure the output does not saturate.  

Your circuit has a gain of +3.2 so input voltage has to be within the range of +/-0.71875V or 1.4375Vp-p, which will yield the full output range, so the input range is not limiting.  

You can use pretty much any op-amp on a single power supply provided you have enough supply voltage and provided you bias the input within the working range and provided you keep in mind the available output range. 

In general for a low power circuit you would want to use higher value resistors than you show. You are loading the output with 5K||(2.2K+1K) which is lower than 5K, obviously, so the output swing is not guaranteed. Normally you can go at least 10x higher for the feedback resistors, maybe considerably more. If you can increase the load to 25K or 100K, and increase the feedback resistors by 100:1 it would be better. You may have to add a small capacitor across R3 to assure stability if you go very high with the resistors.