In order to use the SEMrush API, you will need a paid 'Business' subscription.
URL Profiler enables you to pull back SEMrush data in bulk at the domain level:
To access this and enter your API key, check the SEMrush tickbox under 'Domain Level Data'.
This brings up an API box at the top for you to enter your API key (see below for instructions on this) and a dropdown to select which national Google database to use.
SEMrush maintain 26 national databases, so if you are profiling British sites then you would probably choose the google.co.uk option.
If you are profiling URLs from many countries, you could let the tool automatically choose a database based on the domain’s TLD – defaults to the US database if TLD isn’t matched.
Whilst SEMrush do offer a free account, you need to pay to access their API, either through their 'Business' plan or if you set up a 'Custom Plan'.
If you already have a paid subscription, simply head over to this page whilst logged in:
Simply grab the orange API key and paste it into the box on URL Profiler (see above).
Early 2017, SEMrush changed all their API allocations, such that 'Pro' and 'Guru' subscriptions were no longer allocated any API credits - only the 'Business' subscription now offers API access (you can also sign up for a 'Custom Plan' to include API access).
In terms of ongoing subscriptions, SEMrush claim that this is 'case sensitive', so if you previously had API access but no longer have it then I would suggest contacting SEMrush to clarify.