The DH parameterization reduces the full transform between two joints to a transform with 4 parameters. This parameterization will fit many, but not all, possible kinematics. Let's assume that your instructor did not give you an impossible problem, and that your robot kinematics can be mapped to modified DH parameters.
There are 4 parameters between each joint and thus 4 basic transforms between each joint. For kinematics, the important thing is that the transforms take you to the axis (or line) of the next joint. It does not matter where on that line you arrive. By convention we say that the Z axis of each joint coordinate system is the axis we rotate about or translate along.
Imagine the axis/line of each joint to be infinitely long and you must use only the 4 transforms in sequence (Rotation about X, Translation along X, Rotation about Z, Translation along Z) to go from anywhere on one line to anywhere on the next. So the diagram of the DH transforms will often look different than the physical diagram of the robot. For example, it would be common to use parameters from Joint 1 to Joint 2 such that the location of Joint 2 would be where lines a and d cross in your diagram.
When I do this I use my right hand held up with my thumb and first 2 fingers splayed to represent the Z, X, Y axis and move my hand through the air to help imagine the transforms. 4 transforms between each joint.
Once you figure out the sequence of motions you will be able to use your original diagram to find each of the 4 parameters, and you will be able to insert those parameters into the DH-modified matrices, and finally will be able to multiply them all together to arrive at the coordinate of the end effector.