Let me be clear: yes you can do it yourself, yes it likely won't be perfect. If you go overboard, you can and will do permanent damage to your teeth/gums. You will not have any access to the real quality tools, at least in america to my knowledge. You will have to learn what tartar actually feels like to be able to work with it at all, and from what I remember looking up, a good portion of dentists are not comfortable working on their own teeth. Because you're essentially working blind. Think about trying to cut your own hair; it's like that but worse.
If you are using a sonic toothbrush and flossing at least a few times a week, there's very low chance you'll get anything other than slightly swollen/sensitive gums. If you didn't get cavities regularly earlier, it's likely you won't get it now even if you don't visit a dentist for deeper cleaning/ tartar scraping.
I still wouldn't recommend tartar scraping yourself unless you are a curious idiot like me. Genuinely not worth the time, effort, and potential risk, especially since it likely won't be as good as even a half decent dentist. I only did it because... if my old bad dentist could press extremely hard against my teeth with a honed scraping edge twice a year, I figured it shouldn't be an issue if I do it myself with less pressure than that once a year. Even with a tool of questionable quality.
All that said, easiest way to understand what "clean" feels like is to go to a dentist for a cleaning, then feel your teeth. I used my fingernails and floss smoothness as a gauge on my last visit. I know my fingernail should basically glide through wherever I slide it along teeth; the surface or inbetween. If there is friction/resistance to my nail or floss, I'll know tartar has built up. And it can build up in as soon as a couple weeks really.
You can feel it with the scraping hook tool too. Though that is definitely an experience-only thing.
The general method is to insert the base of the cutting edge hook at the bottom of your tooth between gums (assuming inbetween teeth), then lifting up with the honed/cutting edge pressing against the tooth in question. Of course, do this on parts where you can feel/ know where tartar is. I recall watching videos of that in action, so if you are going to try, look at an example video first. It's very likely you don't have an extreme case like any showcase video.
While in this chapter a sonic scraper is used, and I have no doubt that's good, do remember that is a lot of vibrating metal against your teeth. I really wouldn't use one on yourself at home, as that is probably far more dubiously made for whatever is sold to the general public versus a dental professional. And well, if a sonic toothbrush is 10s of thousands of brushes per minute, thats 10s of thousands of scrapes per minute against your tooth if you aren't hitting tartar. And that does sound very damaging.
If you do try yourself, err on caution side to start. A dental cleaning is ultimately not that expensive, with or without insurance. If it is though, get yourself an electric/sonic toothbrush and at least use that regularly.
Lastly: If you are trying to save yourself from any sort of ongoing tooth pain, it's too late. You absolutely need to see a dentist about the cavity or whatever issue is with your tooth/teeth. And yes, it's likely going to be pretty costly. After fixing the issue, practice good oral hygeine going forward. And again, electric/sonic toothbrush goes a long way.