I'm planning to using this existing outlet to add an additional outlet for a TV directly above. The kit I purchased has a plastic old work receptacle box. Can I use building wire (Romex) into this metal box to connect to the new box? And how do I ground the outgoing cable - can I ground it directly to this metal box using a pigtail?
You can't use Romex. You need AC cable.
Your jurisdiction does not allow the use of Romex cable for electrical wiring. Romex is a brand name of a type of plastic cable called NM-B.
I'm well aware that the Internet is full of instructables and videos that say to use Romex. This is a cheaper wiring method that is not allowed in your town, for reasons.
This is obvious from the builder's use of AC type armored cable and backstabs. The AC type with steel boxes is a deluxe wiring method the builder would not have used if it wasn't compelled by law.
If you have any doubts, go talk to your city's permit issuing office and ask them about wiring requirements there.
But I only know Romex, I don't know anything about AC cable
It's not that big a deal. You use AC cable obviously. The cable clamps appear to be built in to the existing metal box. You'll need to make a bottom entry because all the top holes are full, then loop upward.
So you'll need to get a metal box that has AC cable clamps built in. The local electrical supply house can help. If you bought from the local big-box, you can try there but their advice is terrible, so don't rely on clerks. If you wonder why they're consistently terrible, it's because electricians and electrical supply houses constantly troll the big-box stores looking for competent clerks, and hire them away.
We can't tell from just a picture, but because the existing installation didn't use it then it probably isn't allowed. You haven't provided building type and occupancy to make an informed answer.
Below is a summary allowed use of Non-Metallic Sheathed Cable (often referred to by a brand name Romex), not application of all 800 pages of the code. You can view the NEC yourself at https://www.nfpa.org/codes-and-standards/all-codes-and-standards/list-of-codes-and-standards/detail?code=70 .
2017 NEC 334.12 says not allowed in any structure not permitted in 334.10 (1),(2), and (3).
334.10(1) One and two family dwellings and associated structures
334.10(2) Multi-family dwellings of Types III,IV,and V construction
334.10(3) Other structures of Types III,IV,V
"Informational Note No.1: Types of building construction and occupancy classifications are defined in NFPA220-2015..or the applicable building code..."
NM (Romex) is no good in your building
NEC 334.12 point 1:
334.12 Uses Not Permitted.
(A) Types NM, NMC, and NMS. Types NM, NMC, and NMS cables shall not be permitted as follows:
(1) In any dwelling or structure not specifically permitted in 334.10(1), (2), (3), and (5)
prohibits the use of NM in any building type or situation that's not expressly listed in NEC 334.10:
334.10 Uses Permitted. Type NM, Type NMC, and Type NMS cables shall be permitted to be used in the following, except as prohibited in 334.12:
(1) One- and two-family dwellings and their attached or detached garages, and their storage buildings.
(2) Multi-family dwellings permitted to be of Types III, IV, and V construction.
(3) Other structures permitted to be of Types III, IV, and V construction. Cables shall be concealed within walls, floors, or ceilings that provide a thermal barrier of mate‐ rial that has at least a 15-minute finish rating as identified in listings of fire-rated assemblies.
Informational Note No. 1: Types of building construction and occupancy classifications are defined in NFPA 220-2015, Standard on Types of Building Construction, or the applicable building code, or both.
Informational Note No. 2: See Informative Annex E for determination of building types [NFPA 220, Table 3-1].
(4) Cable trays in structures permitted to be Types III, IV, or V where the cables are identified for the use.
Informational Note: See 310.15(A)(3) for temperature limitation of conductors.
(5) Types I and II construction where installed within raceways permitted to be installed in Types I and II construction.
For most posters here, this doesn't matter much, since houses are covered by 334.10 point 1, and most small apartment buildings and such are covered by 334.10 point 2, as typical wooden frame construction is Type V in the building code nomenclature system.
However, taller/heavier-built apartment and condo buildings, like yours, use Type I fire-resistive masonry construction in this day and age by and large, and as a result, cannot be wired using NM cable. While some local codes require conduit for these jobs, others permit the use of metal-sheathed cables such as type AC and type MC, hence what you see here.
So, what should you do instead?
What I would do in this situation is use either a type MCI-A (preferred) or type AC (if you can't find type MCI-A, as it's still kinda newfangled) cable in 12/2 to make the run from the existing box to a new box for your TV receptacle. Note that the new box will need to be a metal old work box, preferably one with integral clamps for type MC or AC cables, in order to ground the new receptacle correctly.
Furthermore, you'll need to pry out or punch out a 1/2" trade size (7/8" actual diameter) knockout either in the top or a free side of the box, then use a type AC/MC connector (straight for a top feed, 90° aka right-angle for a side feed) to bring the cable into the box. The good news is that since the existing wire connectors have a spare hole in them, you can just plug the new hot and neutral into the existing connectors (with the breaker turned off, of course).
Once everything is wired up correctly, you can install a receptacle into the new box (preferably a self-grounding type receptacle as those can ground to the box, otherwise you'll need to use a ground pigtail to a 10-32 screw into the back of the box to ground the receptacle), put a faceplate on, turn the breaker back on, and plug your TV in there.
Lucky for you, the person that used those wire connectors thought ahead and used 4-slots. So, yes, all you have to do is cut out the area for the new box and drop a piece of romex from the new box to the existing box. Then, connect the black wire to the connector with the blacks, the white to the whites, and as for the ground, I would add a 3 slot wire connector and pigtail the ground (run the green into one slot, run a short length of wire from the connector to the existing outlet, and connect the ground from the new romex to the last slot).
Plastic boxes do not have to be grounded, you will simply connect the ground in the new box to the receptacle.
Edit- I would also consider using the screw terminals on the sides of the receptacle rather than the push in terminals on the back. The black wire goes to the brass screw, the white to the silver.
If you're in areas like Chicago or nearby suburbs, you might be required to use Armored cable or conduit. If your building uses metal studs, you might also be required to use armored.
If you're throwing caution to the wind and using NM anyways, I'll answer the second part
And how do I ground the outgoing cable - can I ground it directly to this metal box using a pigtail?
Your best bet is to drill a hole and add a grounding screw. Wrap the ground wire around the screw and then attach it to the outlet. Because the lip is folded, a grounding clip might be hard to attach.