As long as you retain your US citizenship, you must file a US tax return and report all your world-wide income and pay US tax on it. You can claim a deduction or credit for foreign taxes paid that reduces the impact of double taxation.
If you are overseas at least 330 days of the year, you can claim the foreign earned income exclusion and not pay any income tax on income earned overseas. You still have to file a US tax return to report the income and claim the exclusion, and any unearned income (investments, interest etc) is still subject to US tax.
https://www.irs.gov/individuals/international-taxpayers/foreign-earned-income-exclusion
Once you move out of Alaska and establish a permanent residence overseas, you would not be subject to Alaska state tax since you would no longer be a resident of Alaska. The key is that you are domiciled in Cyprus. Your domicile is the place you call home. It is the place where you have significant family, social, and business ties. Such as, your pharmacy, your doctor, your attorney, your voter registration, your library card, your favorite mechanic, your church, and so on. Even if you lease an apartment or buy a house in Cyprus, you might still be domiciled in Alaska, if you maintain most of your important ties there and plan to someday return. Domicile is a tricky concept and there are few black letter rules that say where you are domiciled, it is a matter of your individual facts and circumstances.
Anyway, if you establish your domicile in Cyprus, then you aren't an Alaska resident any more and don't need to file an Alaska return. If you keep your domicile in Alaska, then you still probably owe state income tax, although you would have to review the rules on their web site.
(In fact, I guess Alaska doesn't have personal income tax, so it doesn't really matter. But the domicile rule would matter to other people reading this post who live in other states. Your domicile and state residency might also impact your eligibility for permanent fund payments. You will have to research this yourself or ask a local accountant.)
The IRS really doesn't care about your Visa status in Europe -- if you are a US citizen and you have income, you pay US tax on it regardless of whether or not you follow local laws.
