Depending on the kind of check you want done it is not a simple process, although fairly straightforward.
Most healthcare workers need an "Enhanced" check, which includes Police records and a check of the "barred" list.
As Laine5 says you can ask the carer to apply for their own. However this is only a "Basic" check and will only show any convictions/cautions that are "unspent".
Unless you do 100 or more applications a year (I assume you do not
) you are not able to apply to the DBS yourself you have to go through an umbrella company. Some allow on line applications but that gets more complicated as ID documents have to be authenticated first.
Full details on the DBS and a list of umbrella companies in your area can be found here...The enhanced check cost £44 and then whatever the umbrella company charge normally between £10-£20.
https://www.gov.uk/disclosure-barring-service-check/overview
Hope that helps and good luck