The most important Hawaii background check resource is the Criminal Justice Data Center. You can use their eCrim website to do online searches of anyone's criminal history for a $15 fee (paid via credit card through the website). You can see your results right away, or also order a certified version for a further cost. The state also maintains an online sex offender database that you can search for no charge.
One Hawaii background check resource that isn't going to be very helpful is the Vital Records office, unless you are looking to a family member's background. All vital records (birth, death, marriage, divorce) are limited to relatives only. If that applies, then you can send an application form and $10 fee for each record. It can take several weeks to get any documents back though.