Suppose I want to write a covered call on my 100 shares of AAPL that I have held for over a year. If I sell a covered call of AAPL 125 in December 2015 with a June 2016 expiration, and the premium on those 100 shares is $600, and the option gets called away or expires in 2016, do I owe any taxes in the current year, 2015? If it gets called in say April 2016, is the premium+profit+dividends all long term capital gains for the year 2016?
You owe no tax on the option transaction in 2015 in this case. How you ultimately get taxed depends on how you dispose of the position. If it expires, then you will have a short-term capital gain on the option position at expiration. If it is exercised, then the option is "gone" for tax purposes and your basis in the underlying is adjusted. From IRS Publication 550:
If a call you write is exercised and you sell the underlying stock, increase your amount realized on the sale of the stock by the amount you received for the call when figuring your gain or loss. The gain or loss is long term or short term depending on your holding period of the stock.
In your case, this will be a long-term capital gain.
For completeness, if you buy to cover the option back from the market before expiration or exercise, then it is also a short-term capital gain.
Also, keep in mind that this all assumes that this covered call is "qualified" so that it does not count as a straddle. You can find more about that in Pub 550.
All of this is for US tax purposes.
You would not owe any taxes in the 2015 year, unless you got exercised and called away in 2015. The premium would be short term capital gains barring some other exception I'm not aware of, and if you retain a gain on the underlying shares then that would still be long term capital gains.
If it gets called in say April 2016, is the premium+profit+dividends all long term capital gains for the year 2016?
The profits are long term capital gains and the premium serves to lower your cost basis, dividends have their own conditions so you'll have to do separate research on that, fortunately they'll likely be negligible compared to the potential capital gains and options premium.