While it will depend somewhat on what part of the Victorian era, the answer is "badly." There were periodic efforts at housing reform in the late 19th century, but in the 1840s-50s, as the city's population surged from immigration, New York's slums easily rivaled Dickensian London. Dickens visited and commented on them himself, in fact. In American Notes (1842), he describes entering the infamous Five Points slum:

A kind of square of leprous houses, some of which are attainable only by crazy wooden stairs without. What lies beyond this tottering flight of steps, that creak beneath our tread?—a miserable room, lighted by one dim candle, and destitute of all comfort, save that which may be hidden in a wretched bed...

Mount up these other stairs with no less caution (there are traps and pitfalls here, for those who are not so well escorted as ourselves) into the housetop; where the bare beams and rafters meet overhead, and calm night looks down through the crevices in the roof. Open the door of one of these cramped hutches full of sleeping negroes. Pah! They have a charcoal fire within; there is a smell of singeing clothes, or flesh, so close they gather round the brazier...

Here too are lanes and alleys, paved with mud knee-deep, underground chambers, where they dance and game;... ruined houses, open to the street, whence, through wide gaps in the walls, other ruins loom upon the eye, as though the world of vice and misery had nothing else to show: hideous tenements which take their name from robbery and murder: all that is loathsome, drooping, and decayed is here.

With no regulations to speak of, housing could be found in almost any part of any type of structure. Former factories and old single-family homes were divided up and converted into boarding houses. A typical purpose-built tenement would be 4-5 floors with two apartments per floor. The worst accommodations were found in basements or in the "rear houses," buildings where additional tenants were crammed on the back part of a lot with no street frontage.

Even in the main building conditions were squalid. In a two-room apartment families regularly doubled- or tripled-up and took in lodgers to help pay rent. Usually only one room would have any windows or source of ventilation. Heat, assuming one could afford coal or firewood, would be provided by a stove and lighting from candles or kerosene lamps. (I wrote a little more about heat here.) Bathroom facilities were unsurprisingly pretty horrifying by any standard. Generally an entire building would share a privy in the basement or rear outhouse, meaning anyone living in those areas suffered additional indignities from the overflow and stench.

Lest we assume Dickens was embellishing for effect, in 1865 the Council of Hygiene and Public Health issued a highly detailed study, full of statistics, that analyzed every block of the city and largely confirms the sub-human conditions of the worst sections. I'll re-use a quote from the study that I've used in a another answer before:

The privy, from necessity, is located in close proximity to the rear house, either immediately in front of a window, or just at the entrance to the cellar. [Overflowing] frequently happens after a hard rain... spreading the contents not only over the yard, but in some instances into the cellar, the bottom of which becomes covered with this semi-liquid filth. In other instances the seat and floor of the privy become soiled and filthy to such an extent that they are wholly unfit for use, and the poor tenants are compelled to resort to their chamber utensils, the contents of which are emptied into the garbage-box, into the already overfilled privy, or into the narrow space between the rear walls of the two houses. This space is so narrow in a majority of instances as to render it impossible for a man to pass between the walls, often becomes the receptacle for all rubbish, garbage, and filth of every description, creating an odor so offensive that it is necessary to keep windows closed, depriving the tenants of that source of ventilation.

To get an idea of what a typical tenement block would look like, the report contains an illustration of a newly constructed set of tenement buildings in what is today the Manhattan neighborhood of Chelsea. While much of the worst buildings in this area have since been razed, some blocks still feature mid 19th century tenement-style buildings (presumably some of the studier built), like these on 29th Street.

In 1879, thanks to attention drawn to the issue by reformers and by studies like the one referenced above, a law was passed prohibiting windowless interior rooms in new construction. Thus, a popular "dumbbell" shape began to appear in buildings across the city where a small air shaft nominally let air and light into interior rooms. These still stand in many residential neighborhoods today and the shafts between the buildings can easily be picked out from at a satellite view, like on this block of East 4th Street.

The law was mostly toothless did essentially nothing for existing buildings however, so another attempt went through in 1901 which outlawed the dumbbell's air shafts (which in practice became de facto trash chutes) and required much better ventilation. At this point the dumbbell-style buildings became known as "old law" and new buildings "new law" (and pre-1879 ones "pre-law"). The early 20th century saw fast construction of these larger "new law" buildings, oftentimes on street corners and featuring courtyards. These appeared as much in the growing outer boroughs as in Manhattan, like this one in Brownsville, Brooklyn. Perhaps this is the type of building you picture when you mention New York's larger scale? Happy to expand on any period where I can.

Main sources are Gotham by Edwin Burrows and Mike Wallace, and Greater Gotham by Mike Wallace.