You should be able to get a printout of your unemployment history by visiting the website of your local benefits office.
The site I looked at for an example was the Pennsylvania website which states that you can get a printout between Wednesday and Friday. So you might want to bear in mind that you may not be able to access the information every day.
If this method doesn't work for you I would recommend calling or visiting the office dealing with your claim, and asking for their advice. I am sure they will be able to provide you with the information you need, but different states (and indeed, countries) will have different methods.
Most public Service Agencies will accept a printout from your pc for eligibility purposes but if there is a problem, I think that the benefit agency will be able to send a hard copy direct to the company requesting the information from you.
