Given any fractional number in any base, for example $0.5_{10} = 0.8_{16}$, pretty simple as this gives no recurring fractional numbers.
However, given $0.255_{10} = \overline{\text{4147AE}_{_{16}}}$ is a conversion that has a recurring fractional number.
$${0.255_{_{10}}\times16_{_{10}}}=0.08_{_{10}}\text{ remainder }\color{red}{\overline{\color{green}{4}}}_{_{16}}\\ {0.08_{_{10}}\times16_{_{10}}}=0.28_{_{10}}\text{ remainder }\color{red}{\overline{\color{green}{1}}}_{_{16}}\\ {0.28_{_{10}}\times16_{_{10}}}=0.48_{_{10}}\text{ remainder }\color{red}{\overline{\color{green}{4}}}_{_{16}}\\ {0.48_{_{10}}\times16_{_{10}}}=0.68_{_{10}}\text{ remainder }\color{red}{\overline{\color{green}{7}}}_{_{16}}\\ {0.68_{_{10}}\times16_{_{10}}}=0.88_{_{10}}\text{ remainder }\color{red}{\overline{\color{green}{a}}}_{_{16}}\\ {0.88_{_{10}}\times16_{_{10}}}=0.08_{_{10}}\text{ remainder }\color{red}{\overline{\color{green}{e}}}_{_{16}}$$
Is the above the correct way to display this, and if so, what would be the simplest way to explain that they need to take the remainders from the top and place them from the MSB to the LSB (Most Significant Bit, Least Significant Bit).
This is an algorithm that different authors like to format differently. I would prefer something like the following, since all the symbols in it mean exactly what they mean normally:
\begin{align} 0.255_{10} \times 16_{10} &= \color{red}{4}_{16} + 0.08_{10},\\ 0.08_{10} \times 16_{10} &= \color{red}{1}_{16} + 0.28_{10},\\ &\vdots \\ 0.88_{10} \times 16_{10} &= \color{red}{\mathrm{E}}_{16} + 0.08_{10}. \end{align}
Notice that the first $4$ does not repeat; after the last line you get $0.08_{10} \times 16_{10} = \color{red}{1}_{16} + 0.28_{10}$ again. So the answer is actually $$0.255_{10} = 0.4\overline{\mathrm{147AE}}_{16}.$$
The "repeating digit" bar should not cover any digits that do not repeat, and I think it is better if it is shown only above the repeating digits, not above the subscript that indicates the base. I think it's also bad form to omit the decimal point (which is still called "decimal point" out of habit and custom even though the base is not decimal), and it makes just as sense to write a leading $0$ before the decimal point in non-decimal bases as in a decimal numeral.