Even when I am brewing at somebody else's house, I still generally get my tea right simply by tasting it. It seems like this approach is often under rated in favor of using a thermometer and precise time. Also, you generally don't need any special tea thermometer to take the temperature as a simple meat thermometer works just as well. (It says do not use in water, however, it is quite tolerable to insert the stick into a pot of water or a kettle considering most of a roasting animal is water anyway. You simply should not submerge it as they are not airtight and they use a spring coil to accurately read.)
For black teas, there's usually not a problem of going too hot although I like to brew mine starting at 85 degrees centigrade for 5-7 minutes. This doesn't result in weak or bitter tea contrary to what some people would have you believe. In general the tastetest seems to be the most reliable, but even a strong-tasting tea can be severely weakened by flavorings such as sugar and milk. Consider also that the temperature when you are drinking it may also have an affect, the bitter and astringent properties becoming stronger when cold and sweeter when warm or hot.
When commercial companies brew milk teas, they use a blend of whole milk, lots of sugar and black tea. In the case that skim milk is used, the whole drink tastes watered down, even when the tea is strong and with whole milk they still use Very strong black tea.
If you're drinking the tea straight however, and it still seems weak, there are other factors to consider. Besides the temperature and duration, the type of tea should also be taken into account. First, if you are using loose-leaf, you should know which type it is. For black tea this is usually broken down into Orange Pekoe (OP, full leaf), Broken Orange Pekoe (BOP, broken leaves, usually around 1/4" pieces) and dust, which is what is too small to be put into BOP or is left over after the breaking has been done.
If your tea is OP, it will need a longer brewing time and hotter temperature to take full advantage of its flavor as the full leaves provide less access to both the xylem/phloem and its overall lower surface area. Next would be BOP tea and last would be dust. Also, consider the grade of the tea besides its physical condition. Is it tippy? flowery? Flowery could mean it's not intended to have a strong flavor and in most cases is expected to be consumed straight. Tippy also could mean this but is a less consistent indicator as it merely indicates the presence of leaftips in the batch.
In the case that you're using a tea bag, it's hard to do anything wrong, so I'm assuming it's not that, but if it is, God help you.
Hopefully this information will be helpful to you in your teamaking endeavors.