Checking maximum current provided by charger, how?
To find the maximum current a power supply can provide, check its nameplate.  In your case it says 800 mA at 5 VDC.  There is your answer.

Unless you think the manufacturer was being deliberately fraudulent, (a legitimate concern for no-name supplies especially from the far east), then you have to do your own measurements.  However, you won't be able to determine the supply's maximum current. You can't verify that it meets its specs without a lot of fancy testing, only that it doesn't fail them.  Put another way, with simple tests you can't prove compliance, only prove or fail to prove non-compliance.

To check that the supply can do what it say at one set of environmental conditions, give it the maximum load it is supposed to be able to handle and check the voltage and ripple.  From Ohm's law, 5V / 800mA = 6.25&Omega;.  This is the resistance you should put accross the supply while verifying it keeps its voltage within spec.  Note that this will dissipate 4 W if the supply works correctly.  Make sure to size the resistor accordingly.

If you're not willing to go out and get a 6.25&Omega; 5W (always leave a little margin) resistor just for this purpose, you can build one up from a bunch of other resistors in various parallel and series combinations.  That helps spread out the power, so a few of the right 2 W resistors of common and standard values should get you close enough.

Once the supply is loaded at its maximum current, check both the average DC voltage and the AC component of the output.  This can be done with separate multimeter readings on the DC and AC settings.  The DC reading should be within spec.  If only the nominal voltage was given without a error tolerance, as is common on namplates, then figure anything more than 3% error is suspect for a regulated supply.  More than 100 mVpp ripple for a regulated supply is also suspect unless you have specific specs.  Note that 100 mVpp is a lot less when reported as averaged absolute value with a RMS fudge factor thrown in, as your multimeter probably does on its "AC" setting.  For example, a 100 mVpp sine wave would probably read as 35 mV "AC", and rather less when the ripple is sharp spikes instead of a sine.  A oscilloscope is the best way to really measure this, but with a AC voltmeter you can get some idea and also compare the value with small load and large load to see if there is significant increase.