Measurement of signals off-air (ie. using an antenna to pick up the signal so you can measure the power) is actually
very difficult to do precisely. So, you have to set your goal to only getting an approximate measurement, I think. Something within +/- 10 dB is easier and may give you what you want. And for that, a WiFi sniffer app on a PC or phone is good enough, or perhaps a calibrated WiFi locator gadget. If you must have a precise measurement, hire an engineer who specializes in this sort of thing, and be prepared for answers that are somewhat unsatisfying. Even using good test equipment, the answer will be statistical in nature. By this, I mean that the answer is a graph or table showing that your signal has a probability X of having a signal strength Y. This is because the signal strength will probably not be static in nature, or in other words, the power will be jumping around depending on what is moving nearby. But realistically, not many people need that kind of precision.
To give it your best shot, in hopes of something closer to +/- 5dB, use a spectrum analyzer set up to measure channel power in either a 20MHz or 40MHz channel (depending on your WiFi channel width selections), and set the top line reference to 0 dBm. In this case you will probably be able to measure power down to -60 dBm which is still pretty strong. For lower powers, you can adjust the spectrum analyzer vertical range downwards, but you won't be able to get much lower than about -80 dBm without a more complicated setup (like with an external preamp for example). Set your frequency to the channel frequency used by your WiFi setup. Attach an antenna suitable for the frequency range, either 2.4 GHz or 5GHz band. Here's one you can make for 2.4GHz:
https://mbharris.co.uk/articles/antenna/omni/quarter/
A simple antenna like this must be held underneath only, and preferably with a stick or something taped to the cable to keep the antenna up and away from your body at least two feet or so. Alternatively, you can tape the antenna to a camera tripod. For approximate measurements, you can assume that the reading on your spectrum analyzer is pretty close to the signal received at the antenna (in other words, I am removing the complication of knowing the antenna gain and cable loss in this case to keep things simple). Just be sure to use a good coax cable (ie. not RG59 or RG59 or RG6) and keep the cable shorter than 6 feet long.
Using this kind of setup will give you a feeling for how much and how fast the signal jumps around, which you can then do some mental averaging to get a final number. Your spectrum analyzer update speed has some bearing on how well you can do this so you may have to fiddle around to find the best settings. Beware that the radio signals are sensitive to their environment, as in, they reflect off anything conductive. Doing measurements indoors means that you may get frustrated trying to measure levels and measurements may not be repeatable.
With a simple antenna like the one above, you won't be able to isolate the direction that the signal is coming from. You mentioned backscatter, which implies that you want to know about direction of signals. For that, it would be necessary to use a directional antenna. There are many plans out there. I found a few using the keywords "DIY directional wifi antenna". These antennas tend to be bigger, which makes them less useful for finding strength in a small space, like a room in a house.