How to find the pin map for the LCD 1602 I2C adapter module?
The ￼tronixstuff page about PCF8574-based backpacks for LCD modules and Arduino (with material that also appears in an instructable) includes a sketch that most likely will work with that backpack.

Once you've identified which I2C library to use, and the appropriate object instantiation, most of the general API description (that is, “the commands to deal with this module”) as shown at arduino.cc's LCDi2c Library page and at arduino.cc's LCD API 1.0 page will apply.

The I2C library-include and the appropriate object instantiation for the backpack shown in your link probably will look much like:

#include &lt;LiquidCrystal_I2C.h&gt;
...
LiquidCrystal_I2C lcd(0x27,2,1,0,4,5,6,7);


For documentation of what the numbers mean, see lines 82 to 90 of LiquidCrystal_I2C.h:

@param      lcd_Addr[in] I2C address of the IO expansion module. [...]
@param      En[in] LCD En (Enable) pin connected to the IO extender module
@param      Rw[in] LCD Rw (Read/write) pin connected to the IO extender module
@param      Rs[in] LCD Rs (Reset) pin connected to the IO extender module
@param      d4[in] LCD data 0 pin map on IO extender module
@param      d5[in] LCD data 1 pin map on IO extender module
@param      d6[in] LCD data 2 pin map on IO extender module
@param      d7[in] LCD data 3 pin map on IO extender module


Briefly, 0x27 is the I2C address of the PCF8574 chip on the backpack.  Note, if your backpack has a PCF8574A, its address is 0x3F.

The other seven numbers describe which PCF8574 output pin is connected for each designated purpose.  Most PCF8574/PCF8574A I2C backpacks are wired the same and numbers 2,1,0,4,5,6,7 are appropriate.



Edit 1: If you write an LCD-with-backpack library of your own and expect to publish it, it is reasonable to comply with the LCD API mentioned earlier.  By using standard interface functions, you will make it possible for users to plug in your library in place of others with the same interface.  In addition, the availability of sketches using the LCD API provides you with a range of test material to verify your drivers.

In writing an LCD-with-backpack library, start by solidifying your understanding of each of the parts – LCD and backpack – separately. 

LCD:  Get some HD44780-based LCD units; hook one up using direct wiring or a shield rather than a backpack; install a known-good library and sketch; run tests to verify the wiring.  With the wiring known to work, develop an LCD control library.  This may require detailed study of the Hitachi HD44780U data sheet or other HD44780 data sheets.  The necessary timing for LCD control is picky enough that it would be a fool's errand to not verify the wiring using known-good software before writing your own.

Backpack:  Plug the 16 pins of a backpack into a breadboard, and attach LED cathodes to all the breadboard pins that connect to PCF8574 output pins.  Use appropriate resistors to V+ on the anode sides of the LEDs.  Connect the backpack's SDA, SCL, V+, and ground to that of your Arduino.  Write simple test software using Wire.h calls to send data to the PCF8574, verifying proper operation using the LEDs.  This may require detailed study of the Texas Instruments PCF8574 Remote 8-Bit I/O Expander for I2C Bus data sheet.  Write your PCF8574 control library.  Adapt your LCD control software to send data via the PCF8574 library instead of via direct wiring.