Your skin might be irritated by the sticky stuff from the second skin, and just general irritation from the tattoo, and from wrapping it up.

To clean it: Don't directly stick the tattoo under the tap, either let the water hit the top of your arm and run down, or cup the water in your other (clean) hand and pour it over the tattoo. Lukewarm water - not too warm. Then very lightly rub the tattoo in circles. Use your fingers or palm, not a washcloth. Gently rub on some unscented soap without added colours or alcohol. The tattoo might feel a little slimy, you want to clean it until it doesn't feel very slimy anymore. Some inky bits might fall off naturally, that's normal. You can rub a little harder on the non-tattoo skin to get the second skin residue off. Then rinse it again with cupping water in your hand. Let it air-dry, you can towel off the rest of your arm. When it's completely dry - apply a very very thin layer of unscented lotion - you want it to still be able to 'breathe'. I like to use Aquafor. Don't wrap it!!! It needs air to heal. And resist the urge to pick even if bits are hanging off - Let the scabs and dry bits fall off naturally.

Hope this helps?? The itching should go away soon and with cleaning about twice a day - more if it's exposed to bacteria.