We are not immigration lawyers, so what follows is just what we've gathered from publicly available links regarding working in the United States. Take this for what is worth, free advice. You will want to examine things more closely before making any decisions.
That depends on your qualifications, and your ability to find a school district that needs your skills. There are two issues: H-1B visa requirements, which we can't really give you much help with, and Alaska "certification" to teach.
You will need to be certified in Alaska. All states have their own certification criteria, and process, not just Alaska. All specific questions relating to Alaskan Certification should be directed to the Alaska Department of Education Teacher Certification department.
The State of Alaska's required information for credential review, which would need to be done in order to get certified, can be found here:
The U.S. Citizenship & Immigration Service (CIS) rules say that if the only barrier is a lack of a Social Security Number, which is required in Alaska for for certification, that teachers can be temporarily certified. This is not clear on the Alaska information sheet, but seems to be true in other states. I'd ask Alaska Teacher Certification about this.
In order to get an H-1B visa you would need to find a school district who can and will sponsor you. In addition, the district needs to demonstrate to the State Department of Labor that they can't find a qualified Alaska candidate. This takes awhile, but is not particularly difficult in rural areas of the state, but depends on the specific job opening. The H1-B has specific requirements you'll need to check on. We don't provide advice on this, as we are not immigration experts.
Here are some links for you to do your own research. The private companies that specialize in this sort of thing charge significant fees, and we don't know if they really improve your chances of success in finding a position, or not:
U.S. Citizenship & Immigration Services (CIS) – Official Government Information. Not a private firm. All fees are the officially required amounts, and the processes and information here is always free:
Immigration Support Services – a company that recruits teachers – has some information on their site. I would learn about what is required, and what the costs and time requirements are before contacting such a company. It is quite possible to get hired without paying a private company. We do not endorse this company or their information. There are dozens of other firms that specialize in this sort of thing.