Use an ice pack to cool the site of the infection. Apply a towel-wrapped ice pack for 10 minutes, then remove for 5. You can repeat this as necessary to deal with the itch.
You can also apply a hot compress to the tattoo site for no more than 3 seconds to alleviate the itch. Be sure to test how hot the water is on the inside of your wrist before soaking a towel for your tattoo.
If your tattoo is mostly healed (around 85% of the scabs/sloughing has fallen off), you can take a bath and soak the last bits of skin stuck to the tattoo. Do not do this if you see actual scabs.
If the tattoo is constantly itchy, you may have an allergy/sensitivity to the pigment. If taking an antihistamine reduces the itchy/discomfort of the tattoo, you can assume you have a problem. Talk to your doctor before using any OTC medications.