Yes the simplest microcontrollers can handle this.
The humidity sensor itself works by changing capacitance. The way you work from there is use the capacitance as the timing cap for a 555 and count the number of cycles per sec. Humirel makes bare sensors and published a schematic for a very accurate circuit including temp compensation, and gives a chart of freq vs RH.
Now a full module like the SHT11 has its own ADC for that and communicates via SPI which is easy. But these are a bit more expensive. IIRC a SHT11 has a precision temp sensor too and that's a good thing. Actually getting precision temp on a PIC takes some "work", and most common thermistors lack high precision in this area unless you calibrate it in software.
Note that RH changes with temp, even if no moisture is added or removed from the air. "dry" warm outside air with a low RH will have a much higher RH if it cools down, even without the addition of water. However, the way air conditioning works, the coils are very cold, below the dewpoint, and water condenses on them as it passes and that gets drained away. That's why you don't get wet, mucky inside air after cooling dry outside air.