Henry became a county in 1819, so there is a long history for the area and the vital records on file often will go back that far (or farther). The state of Alabama officially began to require counties to submit their birth and death registrations in 1908 so you will find the collection of records will be most complete after that.
Vital records can be obtained at the health department office for the county. In Henry County, its located in Abbeville. They can get you certified copies of most kinds of Henry County public records though there is a small fee involved for each search requested. The request forms can be picked up at the office, or downloaded from the Alabama Department of Public Health's website. They are fairly simple, and require that you identify the records you seek by name, dates and locations as well as identify yourself. You'll need to show photo ID at the desk when you make your application for any Henry County public records.
If you send your forms by mail, either to the health office or to the state Vital Records Department in Montgomery, you'll need to include your ID as a clear photocopy.
Henry County Birth Records
You are able to make a request for your own Henry County birth record without any restrictions, or a record for your own child, parent, sibling or current spouse. Other that that, you may run into some difficulty due to the 125-year privacy law. Any records from outside your immediate family are restricted for 125 years, at which time they can be requested by anyone. You can also research them at the State Archives after this point. Certified copies at the health department will cost $15 each.
Henry County Death Records
Since you can't forge a new identity from death certificates, the restrictions on Henry County death records are much more lax than for birth records. Recent records are still held only for immediate family members but that stops after just 25 years. Then the information is public and can be found at the State Archives but also on my genealogy websites. If you need a certified copy, the fee is $15 for each record searched.
Henry County Marriage Records
Henry County marriage records are not kept confidential for any period of time and can be requested by any member of the public. The state record of marriage registrations begins in 1936, and the cost to get a certified copy of a marriage certificate is $15. Probate courts have been recording marriage a lot longer than the state, so you an also ask there if you need an older record recovered.
Henry County Criminal Records
Criminal records in Alabama are permanently confidential unless you are an law enforcement official or a registered employer who is permitted access for the purpose of background checks. You can only make a request for your own Henry County criminal records, and your forms should be sent to the Alabama Criminal Justice Information Center (they're in Montgomery). These records are $25 apiece and the ACJIC website has the forms you need.