Milk doesn't do well at the ideal range for brewing coffee -- it starts to smell like cheese as the milk solids cook. Probably the best way to get near where you're trying is to either make an infusion or a cold extract.
I'm not exactly sure what your object is in brewing coffee in milk -- but if you're looking for something like one of the commercially bottled, cold, coffee-milk drinks, your best bed is probably starting with a cold extract. There are very good inexpensive home extract makers such as the Toddy (Toddy Coffee Maker, Toddy Maker, Cold Brew Coffee Maker, Low Acid Coffee, Healthy Coffee
), and some of the expresso chains like the Coffee Bean sell prepared extract.
If, on the other hand, you're trying to make a latte -- you need (a) to start with espresso; and (b) a steamer that can handle the appropriate volume of milk in a short enough time so as not to "cheese" it. This means a pretty good espresso machine -- and they aren't cheap.
There are less expensive workarounds, but they land you in a sort of "better than Starbuck's" territory. If you buy decent beans (which don't have to be expensive, but often are), and brew it carefully, the underlying espresso will be better than Starbuck's. Starbuck's doesn't use particularly good beans, uses a very demanding roast profile they no longer do very well, and isn't very good in the greater scheme of things, so that's a "take if for what it's worth." A lot of people are delighted with Starbuck's espresso, a lot are delighted with barely "better than Starbuck's," and more power to 'em. No judgment is passed on their taste -- there's no truer cliche than de gustibus non disputandum