Solid-state relay realized with bipolar transistor
A typical DC SSR has only two output connections, so the current for the transistor base must come from the load current. 

So, think of configurations that would do that- for example, a triple darlington with the phototransistor at the 'top', or a hybrid MOSFET/BJT output stage with the MOSFET as a driver. 

Edit: 
Here's a somewhat similar idea with less voltage drop than a 3-transistor Darlington: 



The phototransistor causes base current in the PNP transistor which provides base current ot Q3 which switches the load. 

Q1 and R2 form a constant current circuit for the LED in the optoisolator allowing wide input range (4-12V by their numbers). R1 and Z1 and D1 protect the input from transient overvoltage or reverse voltage. D2 protects the output from reverse voltage. 

By using a Sziklai pair  Q2 and Q3 (sometimes called a complimentary Darlington) configuration as the output stage, the voltage drop across Q3 can be used by the phototransistor so the total voltage drop under load is Vce(sat) + Vbe(on).