What is the range of resistance you’re measuring? What measurement uncertainly can you live with?
We have an AEMC 5600 micro-ohmmeter in our lab. You can select excitation currents of 10 mA, 100 mA, 1 A, and 10 A. But the maximum resistance it can measure is 200 Ω.
Standard DMMs use a constant current source of a fairly low value, and the nominal value depends on the range. As an example, here are the resistance ranges and nominal currents for the Keithley 2001 DMM:
Range_____Current
20 Ω_____9.2 mA
200 Ω___0.98 mA
2 kΩ____0.98 mA
20 kΩ_____89 μA
200 kΩ_____7 μA
2 MΩ_____770 nA
20 MΩ_____70 nA
200 MΩ___4.4 nA
1 GΩ_____4.4 nA
So, one easy way to “dial up” a current is to disable the DMM’s auto-ranging and manually select the range. The drawback, of course, is that you will probably not be in the optimal range for the resistance you’re measuring; you will either lose resolution (by being in too high of a range) or it won’t be able to take a reading at all due to over-ranging (by being in too low of a range). OTOH, you can simply use the meter as a constant current source, and then independently measure the resistance of the device under test (DUT) by measuring the voltage across the DUT, along with the actual current through it. Keep in mind, though, that the maximum voltage the meter can produce in a given range is pretty low.
But since you mentioned you want to evaluate self-heating, I doubt a DMM would be suitable, so your best bet is to use a power supply that can be configured to be a programmable constant current source. We have three or four of these in our lab. With these supplies, you program the value of the constant current, and it measures the actual current (which will be very close to the program value). You then measure the resistance of the DUT by measuring the voltage across the DUT.
If you don’t have such a supply, you can try using a regular ol’ constant-voltage power supply (which is more common), and adjust the voltage to get the current you want. But sometimes there are problems with trying to do this. An improvement on this approach is to include a series potentiometer or rheostat, and you adjust it (and the power supply voltage) to get the current you want. The drawback with this approach is that the current won’t be perfectly constant, especially when the value of your resistor goes up or down due to temperature, but the stability might be good enough for what you’re doing. Finding a suitable potentiometer or rheostat might be challenging, too. If you’re going to use this approach, consider using a fixed resistor in series with a potentiometer or rheostat.
At any rate, there are a number of ways to do this. Much of it depends on the equipment you have access to, your budget, and your time. Also keep in mind the dangers of thermal runaway. If the DUT has a positive tempco, thermal runaway could occur if the (constant) current is above a certain value.
BTW: I wrote a paper on how to measure and calculate the temperature coefficient of a resistor. If you’re interested, shoot me a PM that contains your email address, and I will send it to you.