Answer

What does strategy mean? Well we only get to decide whether to keep / stop rolling the dice. So the strategy means we need a rule for when to stop rolling the die. Let's try with an example to see the dynamic. Should we roll the first time at all? Well we have 0 to start with, so we stand $\frac{1}{6}$ chance of rolling a 6 and staying at 0, or with $\frac{5}{6}$ chance we gain some amount. On average, we gain $\frac{5}{6}\cdot \frac{5+4+3+2+1}{5}=\frac{15}{6}$. So our expected change in wealth if we roll is $\frac{1}{6}\cdot 0 + \frac{15}{6}\gt 0$ - so we should roll as we're expected to gain a positive amount of money.

Now there must be a point at which it is no longer worth re-rolling - i.e. we stand to lose more in case we get a 6 than we stand to gain if we get something else. So let's first extrapolate the logic to a general case: when you already earned $X$ amount during the game, re-rolling will change your wealth by $\frac{1}{6}\cdot (-X) + \frac{15}{6}$. You can see that the larger the $X$, the larger the negative term - and at some point once you have enough wealth, the expected change in wealth becomes negative, i.e. the loss weighted by probability $\frac{1}{6}$ is greater than the expected benefit. This happens at $\frac{1}{6}\cdot (-X) + \frac{15}{6}\lt 0$, i.e. when $X\gt 15$. Therefore your strategy is: you keep rolling until your cumulative sum reaches at least 15, and then you stop rolling.