Kn2222A darlington pair + a relay, not functioning at low currents
A darlington transistor configuration cannot turn-on down to anything close to 0V. The first transistor will saturate (according to the data sheet at 0.3 volts) and, as it takes its "power" from the collector of the 2nd transistor, the 2nd transistor's collector will be probably at about 1V when fully activated. This means you are probably only getting 4V across the relay at best.

Hfe is only really guaranteed to be about 50 on the first transistor and possibly about 100 on the 2nd transistor. Read page 2 of the data sheet. But these are at voltages of 10V. Maybe this will be lower at 5V, I can't say for sure.

All this little factoids make me think your expectations of the transistor, when wired as a darlington, are too high. Maybe consider a MOSFET?