How do I make chai latte at home?
Frothed milk is not essential to Masala Chai, it's normally made with plain milk; the frothed-milk version is mostly a Western variation. But I assume you like it with that little extra. You might try using a milk frother, you can get them much cheaper than the deluxe model I've linked to here. If you're very particular about your frothed milk though, it seems from some quite detailed instructions I found, that a proper milk steaming attachment as found on espresso machines is the way to go. The author of the instructions states that “With a little care, you can create steamed milk that is velvety smooth like the texture of wet shaving cream. The bubbles will be so small that you can barely see them! This is the way it’s supposed to be, because this way, it will blend with the espresso, creating a harmony of the flavors instead of a dry, tasteless cap floating on top.” The same author provides instructions on how to make Masala Chai with or without frothed milk:

In its most basic form, chai is black tea that is brewed strong with a combination of spices and is diluted with milk and sugar.
The spices vary from recipe to recipe, but usually consist of cinnamon, cardamom, cloves, pepper and ginger. Chai tea is traditionally consumed hot and sweet. The sweetness is needed to bring out the full flavors of the spices.
How to Make Chai
Here is a simple recipe to make chai yourself at home. This is what you will need:
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Chai Latte
I just love a good chai latte! A chai latte is just the spiced tea mixed with milk that’s steamed from an espresso machine. I love the soft foamy texture the steamed milk adds to the chai.
You can get these at coffee houses, or if you can make them yourself if you have an espresso machine. I have detailed instructions on steaming (or frothing) milk in the coffee section of this site.
You can follow the above directions just don’t add the milk. This makes a strong tea. Then add the steamed milk straight from the espresso machine.
Practice to make velvety steamed milk and use fresh ingredients. This will insure that you make the perfect chai tea latte.
Source: http://www.2basnob.com/chai-tea.html

You may also want to take a look at the page titled “Milk Frother Tips” on coffee-makers-cafe.com, which gives a comparison between different methods of frothing milk. But note that that page also states you will get the best results with an espresso machine's steam wand:

The only way to produce true microfoam is with a good quality steam wand that has enough pressure to swirl the milk and the ability to heat the milk to the correct temperature. You won't get microfoam with a frother. Microfoam is like a thick milk... like runny-yoghurt in consistency. If you swirl it around in the jug it has a shiny, acrylic-like appearance.