This depends on whether you want the general population or the actual fabric of the city. If the later then we can look at some quite extensive sources on the development of the city and the size and scale of it's buildings. However, as the city of Rome went through several stages of development from the Bronze Age settlement on the Palatine to it's height I think one of the most obvious ways to measure the official extent of the city is the pomerium i.e. the official city limit. Defined by the wonderful Gellius as:

"the area marked off from the surrounding fields by the augurs and ringing the entire city; it is behind the walls and limited by fixed boundaries. The pomerium forms the extent of the urban auspices." (Attic Nights 13.14.1-4)

This way we can get a reasonable accurate picture of the extent of the city in a way that can actually be plotted on a map. (for more on the pomerium see here)

To the ancient sources!

They can give something of an indication of the city size and extent, for example look at this from Livy (History 1.44.3):

"[The census report undertaken by the king Servius in the C6 BC reported that 80,000 (sic) citizens lived in the city, many of them newcomers.] To address the needs of this population, it was clearly necessary to expand the city. Servius added two more hills—the Quirinal and the Viminal—and then enlarged the enclosed area of the Esquiline as well, where he himself took up residence to lend this quarter some status. He surrounded the city with rampart, trench, and wall, thus extending the pomerium."

So from this we can deduce that the city was a) already pretty big, 80,000 people. B) Getting bigger and C) That the leaders were having to respond to the expanding city by officially incorporating more areas into the pomerium.

But

As with anything in ancient history, it is far from simple, (histiography has a way of catching up with everyone). As Dionysius point out, the size of the city was dependent on perspective:

"Today [c. 20 BC] the homes of the city spread far beyond the walls, unprotected and vulnerable to attack, should an enemy come. Indeed, the extent of the city is deceptive for any observer trying to determine where it begins and where it ends, since the urban area is closely intertwined with the countryside around it and gives the impression that the city stretches on forever. If however you judge the size of the city from the circumference of the walls (not an easy thing to do, since buildings are now incorporated into the walls for much of their course, leaving visible however some traces of their ancient structure), Rome would appear to be not much larger than the walled section of Athens."

The defensive walls of the city did not necessarily co-inside with the official city limits.

Technically, tradition dictated that only a leader that had extended the limits of the empire could extend the actual city limits. During the republic this fell to successful generals, but after Augustus, this became the sole preserve of the Emperor. Tacitus notes that Claudius does just this after the successful conquest of the far-flung island of Britannia:

"Claudius also extended the pomerium of Rome [in AD 49], by an ancient custom whereby those who extended the empire might also expand the boundaries of the city. However, Rome's leaders, even those who greatly expanded the empire, had not availed themselves of this privilege, with the exception of Sulla and the deified Augustus." (Tac. Annals 12.23)

This is backed-up archaeologically by boundary stones dating from the period which proudly proclaim Claudius' actions.

From this map you can see the gradual development of the size of the outer defenses of the city.

But, this doesn't really give much indication as to city density etc. As mentioned above, depending on your perspective on Rome the city could look larger or smaller depending on which areas you're prepared to include. The issue is further complicated by the fact that large areas of the city, such as the Campus Martius or the houses and private horti of the aristocracy were largely public or open space, while poorer areas likely were VERY densely populated.

For the common people, home was likely an insulae i.e. apartment/tenement building, with the better off living above their shop near the ground floor and the poorer nearer the top. There are some examples of the lower levels in Pompeii and Ostia Antica, which likely are very similar to the ones that would have been found in Rome.

This makes things like population density estimates very difficult to actually make (especially as Rome is such a densely packed city today).

There are a few things to bear in mind:

• Rome itself, unlike many of its settlements, developed quite haphazardly, and it wasn't until fires that any form of urban planning would effectively be put into action.

• Even an official census of Rome wouldn't give an absolute figure, and likely excluded slaves, maybe women, and people of dubious nature who wanted to remain under the authorities radar.

I don't have any more time to finish off this, but it's a really facinating part of ancient history.

I'd recommend looking at some of Mary Beard's (Cambridge Classicist) documentaries about Rome where she talks about some of these issues and Roman life in general: http://www.youtube.com/watch?v=oWDkMMUzvqQ

Also this GREAT interactive map of Augustan Rome:

TL;DR : Rome got bigger over time. The pomerium was the official city limit but it's very complicated to get a figure for actual size and population.

EDIT: formatting

conservative estimates (Tertius Chandler) 500 000 - high estimates (George Modelski) 1 200 000

As OTK_BBD has pointed out, there are some issues surrounding "Roman" population in Italy and Rome itself, and scholars are relatively split as to the numbers. By the 1st Century CE, Roman population in Italy has been argued to be as low as 6-8 million by P.A. Brunt and as high as 16 million by E. Lo Cascio. This discrepancy has to do with how we read the census records left to us by the Roman administration; Brunt understands the census numbers to be relatively reliable once we account for under-reporting (a problem we still have today) while Lo Cascio and others in the 'high-count' camp believe that the numbers reflected a small portion of the population (male heads of households), thereby leading the census numbers to be drastically lower than reality.

Both camps have trouble making sense of the sudden jump in the census figures during the Augustan era. The low-camp (Brunt et al) believes that the huge difference in the reported numbers can be attributed to women and children now being counted in the census (attested by Augustus' emphasis on the importance of family, a rather weak case to be sure) as well as to the extension of Roman citizenship after the Social Wars to most of the inhabitants of the Italian Peninsula; this allows the proponents of the low-count to mostly stick with the Roman numbers, after some slight modifications--which we've covered.

The high-count basically assumes that the census figures never cover women and children, and only adult heads of households (possibly not even adult males who were under the political hand of their respective fathers) would have been counted. This allows the high-count to inflate the census numbers by quite a bit, with the final count being anywhere from 14-16 million Roman citizens by the 1st Cent CE.

So, both of these have their issues (agricultural output, urbanization rates, arbitrary modifications in under-enumeration) and thus cannot truly be said to be right or wrong, though my personal belief leans toward the lower count. What these two camps both have in common, however, is the basic assumption that the city of Rome would have likely contained ~15% of the total Roman population. So, based on the numbers which one believes are true, this population could vary widely as well.

BUT! We should also consider a very significant part of the population that is difficult to enumerate in itself--Slaves and Freedmen! The city of Rome, especially, would have been home to a large population of slaves and freedmen. Overall, Brunt claims an approximate 3:8 Ratio (slave/freedmen:Roman citizen) in Italy, which we could probably extrapolate to the city of Rome with little difficulty.

The conclusion to this is that this issue is incredibly complex and will be argued for a long while. I have argued in the past that the population of Rome would have never reached a number greater than 1.5 million with citizens and slaves both included, as the water supply would have directly influenced population growth and sustainability. With all aqueducts operating at full capacity (which likely was rarely if ever the case) the city, in the 1st Cent CE, would have been able to sustain about 1.6 million at 500 liters/day/person after the construction of the Aqua Claudia. However, I have my doubts that the population ever reached much higher than 1.1 million.

Sources: Brunt, P.A. 1971. Italian Manpower: 225 BC-14AD, Oxford

Hodge, A.T. 1992. Roman Aqueducts & Water Supply, London

Lo Cascio, E. 1994. The Size of The Roman Population: Beloch and the Meaning of the Augustan Census Figures.

Morley, N. 2001. "The Transformation of Italy, 225-28 BCE." Journal of Roman Studies 91: 50-62.

Sheidel, W. 2007. "Roman Population Size: The Logic of the Debate" Stanford. (This source in particular is a great introduction to both sides of the issue)

*To be clear, these scholars do not represent the 'champions' of the debate, they are just the few scholars I have had the chance to read through.