Apparently, you can use acrylic paints to decorate your flask, and then seal it with a dishwasher-safe sealant.
For painting you can use regular acrylic paints, and for doodling you can use acrylic paint markers.
I suggest to opt for higher quality products, as they will have a higher amount of pigment and improve the coverage on your flask.
There are paint markers available that are dishwasher safe (like, for example, these ones).
For other products you might have to use a sealant.
In the linked article, the flasks were painted with regular acrylic paints, and afterwards a dishwasher-safe Mod Podge sealant was used to prevent if from peeling and chipping. This seems to work fine, but remember that this particular sealant is not food grade. However, as long as it is not applied to a surface where it will get into contact with food or your lips, this should be a safe choice. You might also be able to find sealants that are both dishwasher-safe and food grade.
Make sure to heed the drying time, though, to guarantee the highest level of inertness.