Measuring T1 and T2 constants on IBM Q

I cannot give you a complete answer(I am not too familiar with the IBM quantum tools) however I might be able to give you a few hints from a NMR/EPR perspective.

In magnetic resonance T2 is commonly measured by generating a spin coherence, and refocusing at progressively longer times then measuring a spin echo.

In quantum gate language that would be: prepare in state $|0rangle$, Hadamard gate, $X$ or $Y$ gate, Hadamard gate, measure. And progressively add identity operations between the $H-X$ and $X-H$. Alternatively you can add progressively more $X$ gates and only measure at the odd numbered ones. Though I suppose the latter method would introduce gate errors into your measurement.

And T1 in magnetic resonance is usually measured by an inversion recovery: so prepare qubit in state $|0rangle$, invert with $X$ or $Y$, and measure at progressively longer delays by using identity operations as delays.

Lets start with measuring circuits. With link to user245427 answer, you should construct following circuits in composer followingly:

T1 (relaxation time)

T1 is constant connected with spontaneous relaxation from state $|1rangle$ to state $|0rangle$. So firstly apply $X$ gate on a qubit to change its state from $|0rangle$ to $|1rangle$, then apply a few $I$ gates to make a delay and then measure results. Record a probability of measuring state $|1rangle$ (i.e. relaxation did not occur).

T2 (dephasing time)

T2 is connected with change in phase, for example state $|+rangle$ changes spontaneously to $|-rangle$. To prepare $|+rangle$ apply $H$ gate on qubit in state $|0rangle$. Then apply several $I$ gates to make a delay. After that apply again $H$ gate and do measurement. Record probability of measuring state $|0rangle$ (i.e. phase change did not occur).

T1 and T2 calculation

Both decoherence processes are described by exponential decay law:

$$ P(t) = mathrm{e}^{-frac{t}{T}} $$

where $T$ is eiher T1 or T2 constant, $t$ is time between setting qubit to either state $|1rangle$ for T1 or $|+rangle$ for T2 and its measurment. Having probabilities that a qubit is in state either $|1rangle$ and $|+rangle$ and knowing $t$ you can easily calculate

$$ T = -frac{t}{ln P(t)}. $$

Although it is not problem to construct such circuits on IBM Q, I realized that the problem is how to obtain time $t$. After simulation you get results with time the simulation actually run on a quantum processor. It seems logical to divide this time with number of shots to get length of one shot and hence $t$. I did so on Melbourne processor but it seems that some other operations take place among shots which lengthen time $t$. As a result you can not get actual time $t$. If you put lenght of simulation to the formula above, resulting T1 is in order of miliseconds which does not make sense.