In addition. Variables that will effect the strength of your brew are the amount of coffee to water. Typically with an immersion brewing process like you're roughly doing, I use 7grams(~1tbsp) per cup. You can increase for more potent coffee. I use 205 degree Fahrenheit water(10seconds off boil) to brew, any higher will extract tastes you don't want, any lower will underextract. Without a filter you should expect sediment in the bottom of your cup.
Something you could try, since you don't have many tools needed to brew coffee, is a cold water extraction. In Costa Rica, they take the coffee beans and either crush them between 2 stones, much like they would mill corn for tortillas, or they would roughly crush the coffee using something like a mortar and pestel or even something like a rolling pin and a hard surface. Basically you just need to crush the bean into a coarse form. Then they would submerge a burlap sack of the ground coffee in water and just remove the bag after 24 hours.
This brewing process removes a lot of the variables that can contribute to make bad coffee. You don't have to worry about water tempurature, even ground coffee, or precise brewing times.
You could try submerging crushed ground coffee in cold water for 24 hours in a small cloth(food safe) sack of some sort. A ratio of 12oz of coffee per gallon(2liters) of water should work well. I've seen people use cotton clothes for similar processes, and even old tshirts(not recommended). Whatever you use make sure its food safe.
This will produce a very strong coffee concentrate, you could then dilute this with hot water or hot milk to heat it.
Where I used to work we had a drink that used cold water coffee extract(Toddy) and steamed it with milk to heat it up. You could try something like this, heating milk on the stove and adding it to the cold water coffee extract. The extract can be kept for up to a week in the fridge.