How can I make a current source that can supply 1uA?
D2 generates a stable voltage, while the forward voltage of Q1 varies with temperature (by about -2 mV/deg. C). You can compensate for this by putting a standard diode in series with D2 -- e.g. a 1N4148. Another transistor like Q2 with its base-collector tied together would also work well.  Then you need to change R1 as the current is then 2.5 V (from D2) divided by R2.
D2 provides a voltage of about 2.495V below the positive rail with 1mA+ flowing through it via R3. That voltage minus Vbe (about 0.6V) appears across R1, so a collector current of almost that much flows,  provide RL is not more than about 2.5M.



simulate this circuit &ndash; Schematic created using CircuitLab
Of course if you need 0.1% precision you'd use an op-amp etc. but that's kind of boring.
And, if you have budget and need something for the lab, something like the Keysight B2901A/2902A will give you 1uA ±(0.025 % + 500 pA) at up to 200V+ for a small fraction of the price of a new car.