It depends on your state's law and your employer's policies. Many states give employees the right to view their personnel files, although there may be documents you aren't entitled to see (such as reports from criminal investigations or reference letters). Some states give employees a right to a copy of at least some of their personnel documents, such as documents they have signed. And even if state law doesn't give you these rights, your employer's policies may allow you to view or copy your file.
For more information, see Nolo's article Your Personnel File and Your Rights.