The NRA social security portal is here:
https://inetdec.nra.bg/pls/pub/home.html
There are some free access services that don't need a password/PIN/QES. In particular, you can check your status, and make contributions (this is pretty much all you need).
If it doesn't work, it's probably because you haven't made any payment yet and/or haven't "activated" your social security. When I first came to Bulgaria I had to go to the Plovdiv NRA office and fill in a little form for this.
Health insurance status
The access is using LN/LNC and your date of birth. LN/LNC is your ID number from your BG residence permit ("lichna karta").
Your status should say: "THE PERSON HAS CONTINUOUS HEALTH INSURANCE RIGHTS". If not, it will show which months are unpaid, and you will need to make some contributions.
Payment of liabilities to the budget electronically
Type of Identifier = LNC
PIN / LNC / BULSTAT / Employee № = your ID number from your card
Liable person = your name (I put in Cyrillic & English, but don't know if it's necessary)
category = NHIF social security contributions (and enter amount)
Then you can pay via the portal or via Epay.
When your social security is current, then you can get a BG EHIC card (via branch of DSK) for use in Europe.
I am not sure whether you need a card issued for a local doctor, or whether he checks if you have insurance using this system. So I haven't figured this out yet. And private treatment is so cheap I haven't had to worry about it yet.
If you're an employee, I don't think you have to worry about this, as your employer should do it. But it's good to check (I found my employer wasn't bothering).
I changed my status (at the NRA office) to "unemployed", and I now pay 26 leva per month. This is not a bad deal to be part of the national health system.