Nowadays most states (including Alaska) have laws mandating inclusion of relevant psychiatric records into the federal database. As such the NICS database contains millions of mental health records.
However, federal laws only bar those with very severe mental illnesses from purchasing guns -- specifically those who have been involuntarily committed to a mental institution, or otherwise declared by a court or other authority as being "mentally defect".
Only a handful of states have additional restrictions, and Alaska isn't one of them. Even in Seattle, you can check yourself into a mental hospital on Monday, then buy a handgun on Tuesday.
In this case it's not clear if the alleged shooter was involuntarily committed after he approached the FBI back in November. Besides, Alaska doesn't require background checks for all gun purchases.
It is interesting that quite a few of these "mass shooters" have had relatively recent FBI / Homeland Security involvement.