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The Moon Comes Over the Tower - Meeting Emily
A major part of this quest actually seems to be just getting it. Emily Ortal is supposed to be outside the Lucky 38 Hotel in New Vegas, and I believe that she's supposed to approach you shortly after one of your early trips out of the Lucky 38 after meeting Mr. House. Unfortunately, it doesn't seem to work out like that too often.
For starters, just try looking for her. She should be at the bottom of the steps to the Lucky 38, close to the gate to the next section of the strip. She's in a Followers of the Apocalypse outfit, so she's pretty easy to spot. If she's not there, you can also try waiting inside the HQ at Camp Golf. A number of people seem to have had luck with that (it appears that it was part of a now removed part of the quest).
I also believe that talking to Julie Farkas after going into the Lucky 38 can help. She should mention that they have an operative in New Vegas who could use your help, and she'll ask you to keep an eye out. If all else fails, try advancing the main quest. I didn't get this until I had already killed House, for example.
Also, note that you have the option to pass a Barter check to potentially bump up your reward (or at least confirm it and get some experience).
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The Moon Comes Over the Tower - Bugging the Lucky 38
If you're lucky enough to actually get the quest from Emily Ortal, you have to walk back into the Lucky 38. In order to plant the bug, walk up to the slightly raised area above the cashier's booth to find a computer terminal hanging out of the wall. You don't even have to pass the Science check. A special context menu should come up when you activate the terminal. Chose to plant the bug to wrap up the quest.
Note that planting the bug doesn't really have any negative impacts. House has strong defenses in place, so the bug will get fried in less than a minute. In fact, I believe you can tell House about the plot and he'll just wish you luck.
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The Moon Comes Over the Tower - The Wrapup
As noted, the hardest part of this quest seems to just be getting it. Once you get inside and plant the bug, you're all set. Talk to Emily Ortal outside and she should thank you, but note the failure of the bug. Emily should hand over some very nice medical items and thank you for your help. You'll also get some reputation toward joining the Followers. If you want to be idolized by the Followers of the Apocalypse, this is a good quest to seek out.
New Vegas is a fairly dense quest hub, with a number of side quests to do for various side factions. I'll cover how to help Emily Ortal spy on Mr. House, how to help a sign maker find his inspiration, how to help a wayward girl in North Vegas and how to break up a gang of cannibals in New Vegas.