Here are two examples of the square and multiply method for $5^{69} \bmod 101$:
$ \begin{matrix} 5^{69} &\equiv& 5 &\cdot &(5^{34})^2 &\equiv & 37 \\ 5^{34} &\equiv& &&(5^{17})^2 &\equiv& 88 &(\equiv -13) \\ 5^{17} &\equiv& 5 &\cdot &(5^8)^2 &\equiv& 54 \\ 5^{8} &\equiv& &&(5^4)^2 &\equiv& 58 \\ 5^{4} &\equiv& &&(5^2)^2 &\equiv& 19 \\ 5^{2} &\equiv& &&(5^1)^2 &\equiv& 25 \\ 5^{1} &\equiv& 5 &\cdot &(1)^2 &\equiv& 5 \end{matrix} $
The computation proceeds by starting with $5^{69}$ and then working downward to create the first two columns, then computing the results from the bottom up. (normally you'd skip the last line; I put it there to clarify the next paragraph)
As a shortcut, the binary representation of $69$ is $1000101_2$; reading the binary digits from left to right tell us the operations to do starting from the value $1$: $0$ says "square" and $1$ says "square and multiply by $5$".
The other way is to compute a list of repeated squares:
$ \begin{matrix} 5^1 &\equiv& 5 \\ 5^2 &\equiv& 25 \\ 5^4 &\equiv& 19 \\ 5^8 &\equiv& 58 \\ 5^{16} &\equiv& 31 \\ 5^{32} &\equiv& 52 \\ 5^{64} &\equiv& 78 \end{matrix} $
Then work out which terms you need to multiply together:
$ 5^{69} \equiv 5^{64 + 4 + 1} \equiv 78 \cdot 19 \cdot 5 \equiv 37 $