Sainsmart 2004 LCD with I2C piggyback controller

Hey Don,
thank you for the quick reply!

Thanks for the Link! I've tried the lib "LiquidCrystal_V1.2.1.zip" - I believe I should use the newest one? (Also I tried the rest - none worked)
To test the Display I used the example from the ZIP file.

#include <Wire.h> 
#include <LiquidCrystal_I2C.h>



#define BACKLIGHT_PIN     13

LiquidCrystal_I2C lcd(0x3F);  // Set the LCD I2C address

//LiquidCrystal_I2C lcd(0x38, BACKLIGHT_PIN, POSITIVE);  // Set the LCD I2C address


// Creat a set of new characters
const uint8_t charBitmap[][8] = {
   { 0xc, 0x12, 0x12, 0xc, 0, 0, 0, 0 },
   { 0x6, 0x9, 0x9, 0x6, 0, 0, 0, 0 },
   { 0x0, 0x6, 0x9, 0x9, 0x6, 0, 0, 0x0 },
   { 0x0, 0xc, 0x12, 0x12, 0xc, 0, 0, 0x0 },
   { 0x0, 0x0, 0xc, 0x12, 0x12, 0xc, 0, 0x0 },
   { 0x0, 0x0, 0x6, 0x9, 0x9, 0x6, 0, 0x0 },
   { 0x0, 0x0, 0x0, 0x6, 0x9, 0x9, 0x6, 0x0 },
   { 0x0, 0x0, 0x0, 0xc, 0x12, 0x12, 0xc, 0x0 }
   
};

void setup()
{
   int charBitmapSize = (sizeof(charBitmap ) / sizeof (charBitmap[0]));

  // Switch on the backlight
  pinMode ( BACKLIGHT_PIN, OUTPUT );
  digitalWrite ( BACKLIGHT_PIN, HIGH );
  
  lcd.begin(16,2);               // initialize the lcd 

   for ( int i = 0; i < charBitmapSize; i++ )
   {
      lcd.createChar ( i, (uint8_t *)charBitmap[i] );
   }

  lcd.home ();                   // go home
  lcd.print("Hello, ARDUINO ");  
  lcd.setCursor ( 0, 1 );        // go to the next line
  lcd.print (" FORUM - fm   ");
  delay ( 1000 );
}

void loop()
{
   lcd.home ();
   // Do a little animation by writing to the same location
   for ( int i = 0; i < 2; i++ )
   {
      for ( int j = 0; j < 16; j++ )
      {
         lcd.print (char(random(7)));
      }
      lcd.setCursor ( 0, 1 );
   }
   delay (200);
}

Does anyone know, which Lib to use? Could someone just make up something very small (just print s.th. on the Display)? I think from there on I could untangle myself - but for now all that shows on the display is the 1. and 3. Line Black for contrast...

Thanks for any help!
-Daniel