Using this as a reference for amount of rubbish generated, averages to about 281.5 kg/person per year.
Our moon has a mass of 7.342×10^22 kg, so doing the math, the current population of earth would need 34.63 billion years in order to produce the amount of rubbish needed to create a moon with the mass of our own. See here for a few more details. Also keep in mind that our moon (or other planetary objects with similar mass/size) won't be able to hold an atmosphere, so having humans living on the Trash Planet (ignoring other issues) would mean it would need to be much larger.
Since your civilization is a space faring one, with populations on a number of different planets, it's logical to assume there's quite a few more of them. Taking that into account, we could calculate the total population needed in order to produce the amount of rubbish needed in a more feasible amount of time. So if say, you wanted to build it in the span of 200 years, you'd need a population of 1.304×10^18. That's about 170 million Earths worth of people. Considering this all takes place in the future, you would assume waste would be somewhat reduce (hopefully recycling and such). You could of course have the future people go in the complete opposite direction (what with rubbish being needed for the Trash Planet construction) and have them produce more rubbish, but even doubling the amount of rubbish per person and doubling the construction time to 400 still means you need roughly 44 million Earths worth of people.
All that aside, making an artificial world also seems somewhat tough, at least according to this thread. While you're not really looking to make a whole planet, making it habitable in any form (even with Moon bases etc) seems unfeasible given the timespan needed to get it to settle down.