To get consistent results, you should have consistent steps. Specifically, I'd pay attention to these four factors:
- Brew ratio
- Brew temperature
- Brew time
- Grind size
Personally, I aim for a brew ratio of 1:15 measured by weight (so I'd add 750 grams of water to 50 grams of coffee). This is mostly because it's easy to calculate and you can of course choose a slightly different ratio if you like a stronger or weaker cup. I'd say anywhere between 1:13 to 1:18 is fine, but you'll want to keep track of the ratio you're using so you can reproduce it consistently.
As for brew temperature, I think you should aim to start as hot as possible. For me that means preheating the brewing vessel with water that's just off the boil (as in, it has stopped boiling by the time I've taken it to the French press, no need to wait). So you fill the French press with hot water just to heat it (I add about 200 mL for an 800 mL beaker) and then you discard it (or use it to heat the cups) just before you add the grinds. The water you use for brewing may be just off the boil as well, but if you took it for preheating like a minute ago (but not much longer) and you don't feel like getting it to a boil again then that's fine too.
As for brew time, I would increase the time to 6-9 minutes. The added brew time should benefit the result. See also the linked video below which recommends 4+5= 9 minutes of total brewing time.
Grind size doesn't have to be super coarse, just not so fine as to clog the filter. In his video, James Hoffmann recommends a medium grind, that's a bit ambiguous but he uses it to steer people away from the often recommended (super) coarse. I'd say, if you keep all the other parameters as I described and the cup is still to weak, then go finer incrementally until you like it keeping all the other factors the same.
I've recommended that linked video to others before, but in hindsight I think he's taking some unnecessary steps. For example, he scoops off some grinds and he doesn't plunge down past the surface of the coffee with the idea that more fines will have sunk to and remain at the bottom and the filter acts as a strainer when you pour it (rather than when you push down). I guess the technique is worth trying if you're still experimenting but it's a bit too much of a hassle for my taste.
When you do plunge down, you may want to keep the filter just above most of the sunk coffee, that way you can pour it easily but you don't disturb the sediment more than you have to.
