google a darlington pair and you'll see that the base(s) are not grounded, as you've drawn it.
Also, you don't need so many stages. A single Darlington will work for 10's of uA.
You can either use a packaged Darlington transistor, or build your own from parts you have lying about.
Make sure the first transistor is higher gain than the second transistor in the pair. For example, a BC547B or C, which can have an Hfe above 400, 500 up to 800... and the second transistor should have higher current handling capability for driving the relay. If it's a small relay, than a 2N2222 is common, it can handle around 500ma. However, keep in mind that the actual choice of transistor is highly dependent on the the load relay you are using, how much current it draws, and what the voltage drops will be.
As poster olsenn said, you will want to use resistors in the relay circuit, to ensure that the current is where you want it and the second transistor is saturated when it is on, so that you have very little power dissipation in the output transistor. When saturated, it is simply a switch and no power dissipation means no heat, and no need for a heat sink. This is important, because you have to make sure not to exceed the second transistor's power rating.
Also, consider your voltage available, as well as the current. The Darlington pair needs above 1.4 volts to turn on (2 Vbe drops). If your source of current is also lower than 1.4 volts, then you should consider another approach, like a complementary pair (NPN-PNP) (needs 1 Vbe drop to turn on) or if your voltage signal is still lower than that, you will need a voltage gain stage first, using a common emitter amplifier.
At this point, a single op-amp starts to look more promising. But you don't need an op-amp unless you also need the included voltage gain it provides.