Serial Print a number of int values

how do I read the value of each pin in the array (a variable)

Using a for loop.

byte somePins[] = { 5, 8, 10, 11, 13 };
const byte pinCount = sizeof(somePins) / sizeof(somePins[0]); // Count the pins
byte pinVals[pinCount];

void loop()
{
   for(byte p=0; p<pinCount; p++)
   {
      pinsVals[p] = digitalRead(somePins[p]);
   }
}

and subsequently print the values?

Not surprisingly, the same way.

Also, how do I get the DHT-22 values into the array since they are a part of the library?

The values are not part of the library. The values are outputs from library methodss, just like the values output by the digitalRead() function.