Current Amplifier Design
It's not clear why you want to use a transistor for this. Yes, a transistor is an amplifier, but maybe not in the sense you're imagining, hence your confusion when it comes to reading the datasheets.

I will give you the circuit you need to use the transistor in this application. I recommend building it, and then discovering through experimentation why it's not a great choice for this application. Primarily, the amplification ratio (called the "gain") is hard to control in an open loop.







simulate this circuit &ndash; Schematic created using CircuitLab

All you need to do is size R1 such that at 5V, enough base current flows to allow 60mA of collector current to flow. Use these two formulas:

$$
i_b = \frac{5V - 0.7V}{R1}
$$
$$
i_c = i_b\beta
$$

where \$i_b\$ is the base current, \$5V\$ is the amplitude of the PWM, \$0.7V\$ is the diode drop across the base-emitter junction, \$i_c\$ is the collector current and \$\beta\$ is the BJT gain.

Re-arranging gives:

$$
R1 = \frac{(5V - 0.7V)\beta}{i_c}
$$

So say your \$\beta\$ is 100, and your desired \$i_c\$ is 60mA, then \$R1\$ would be about \$7200\Omega\$.

When your PWM duty is 100%, the BJT will allow 60mA to flow. When it is at 50%, it will allow 60mA for 50% of the time, and 0mA for the other 50%, giving an average of 30mA.

In theory at least. In practice you might find it a little wilder than that, at which point you might be tempted to use something like an opamp in a current source topology instead.