How can I use 7400-series chips to translate the following 5-bit signals into the desired 8-bit signals?
Typically, unless you recognize some obvious pattern (such as the outputs being a shift or inverse of the inputs), you'd write and reduce an expression for each individual output bit, as a function of all input bits.

The classic paper tool for doing these reductions is the Karnough Map (wikipedia)

This will typically give you an expression as a sum of products, which is to say an OR function following one or more AND functions, potentially with inverters in some places - or if you prefer you can use de Morgan's law to convert to its product-of-sums dual, with the gates in the other order.  Each can be implemented by a combination of AND, OR and inverter gates, or NAND and NOR - in fact you can do the whole thing using only NAND chips or only NOR chips.

Some particular functions may be more efficiently implemented with other special devices, for example the `138 decoder.