Hello, thinking of buying a bike in the US and riding around for a few months, crossing into Canada and Alaska. I want to buy and not rent. Has anyone here had experience with doing this, how to register the bike etc etc.

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What ever, lets go...

Basically most states require residency to register and insure a vehicle. There used to be quite a few that were more lax such as Washington state, and Vermont but I am pretty sure that is finished now. California as far as I know is easy to do this in, and there is a workaround in Montana that involves registering an LLC and this ends up giving you a vehicle that has a lifetime registration on it and doesnt need yearly inspections but is a bit of a hassle and not super cheap depending on the value of the bike you are looking at. BC in canada I am pretty sure allows non residents to register and insure a vehicle so that might be a consideration, and other provinces may also allow this.
Depending on how long you want to ride and what you are planning you might be able to find a foreigner who wants to sell his bike after finishing his trip, and depending on where that vehicle is registered you may be able to legally register it in your name or use it legally with a power of attorney.
This is much more common in latin america, if you you look at the want ads in the forum here you will regularly see american, european, and ozzie bikes for sale. If the bike was already in mexico for example that might end up working out, or just keep your eyes open and see what turns up. Practically speaking I think BC or CA would probably be your best simplest solution.

Greetings from Virginia, USA.

A few years ago I had two Danish friends who did this. There are two main challenges:
1. You will need a local USA address to register the bike. If you already know someone in the states or can befriend someone online, that should work.
2. Insurance. Many insurance companies will only insure you if you have a valid US based drivers license. My friends had to do a lot of digging but finally found an insurer that specialsed in this sort of thing. If you do some research ahead of time, you can probably find something.

My friends spent 5 months here and hit 49 of 50 states. They sold their bikes back to a dealer when it was time to return to Denmark.

Good luck!

In short. It's doable, but depends on state and even county.

As previously mentioned you need a US address. The registration papers and license plate will be mailed to that address. Depending on location, hotel, AirBnB, campground or post office box might work. Some states require yearly inspections, it's better to avoid those. Some states don't require a drivers license to buy and own a vehicle (you do need a license to ride it). Some states have yearly charges for owning a registered vehicle. To mention some locations, I hear that California and Texas are some of the easiest states for foreigner to purchase a vehicle.
If buying from a dealer, note that many dealers add all sorts of charges to the showroom floor sticker price. Always make sure that any agreed on prices are "ride-away" prices.
Insurance has been mentioned.
If you want to sell the bike after riding few months you might face large price drop.

There are many threads about this. Recent ones too.

As far as I know, Arizona seems to be one of the best states to buy and travel with a motorcycle as a non-resident.

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Did some trips.
Rode some bikes.
Fix them for a living.
Can't say anymore.

As you can see, the laws governing motor vehicles here are a patchwork of 50 different states. In Virginia, the process goes like this:

1. Have your address and insurance company lined up
2. Go haggle with the dealership. I suggest buying a good used bike and saving some money (and avoiding some hefty fees such as freight and dealer prep).
3. Provide proof of insurance, and ask for a temporary tag. Wire the payment to the dealer. Recommend that you do the DMV paperwork yourself rather than the dealer. You’ll get your permanent tag and registration quicker and save a few bucks. Dealer will sign over the title and give you the bill of sale and some other paperwork.
4. Things that are specific to Virginia and some other states: 4.15% sales tax (paid at time of purchase), annual safety inspection (dealer should already have done this), county or city property tax (varies widely, currently about 2% of blue book value where I live, paid after you get a bill from the county).
5. Go online and book an appointment with a nearby DMV office. Download the application for title and registration and fill it out. Go to the DMV at the appointed time with all your paperwork and get your permanent tag, registration, and title.
6. Go online and notify the local county or city of your purchase. Also notify them when you sell the bike. Eventually you will get a pro rated property tax bill.

Simple! Note that there is no federal govt involvement with any of this. It’s all state and local.