Muxes, Adders, Comparators, and Gates
The steps I usually follow to find the solution are explained here. (Disclaimer:I am not an expert here.)
1. Derive the input-out relationship
The first step is to understand and express the input-output relationship in some form. You can express this relation in a convenient form. Some of the standard forms are

mathematical expression
truth table
logical expression
any combination of above

Taking your question as an example,
If 011xxxx and 011yyyy be the input digits. The last four bits of these digits represent their binary values.
So xxxxyyyy is the BCD representation of the number. Converting this BCD to binary is the required operation here. And this can be done by adding lower BCD digit with ten times the upper digit. ie,
$$\mathtt{output = 1010\times xxxx + yyyy}$$
So we have obtained the input output relation here. But we don't have a multiplier to implement this relationship.
2. Re-formulate the relation in terms of available hardware
Multiplying with \$1010\$ can be done by shifting and adding as given below:
$$\mathtt{1010\times xxxx  = xxxx000 + xxxx0}$$
OR
$$\mathtt{output = xxxx000 + xxxx0 + yyyy}$$
So here we have obtained the input output relationship in terms of available hardware. A 7-bit adder can do the addition. zero extension units can do the zero-padding.
Pad a zero in front of answer to make it in 2's complement form.
3. Circuit diagram/ block diagram
Once we have the input-output relation, then replacing the operators in the expression with the corresponding block/module will give you the circuit diagram.