First we need to figure out how much we need to dig. To cut costs a little, let's say each person needs the same amount of space as an average person in Hong Kong; 161 square feet. Let's also say (to make the math easier) you have 10 feet heights and that the beast companion just takes up the same amount of space as another person. This means our (fairly small house) needs \$ 161 \times 10 \times 5 = 8050\$ cubic feet carved out.
From the answer to this question we get that properly equipped dwarves can each mine 70 to 35 cubic feet per hour, depending on the hardness of the stone. If we assume hard stone (better structural stability and accounting for structures (like rooms, stairs etc.) they need to make) our small house needs \$8050 / 35 = 230\$ dwarven work hours to excavate.
The services table of page 159 of the Player's Handbook lists the wage of a skilled hireling at 2 gold per day, and assuming 8 work hours per day the cost of excavating the house becomes \$2\times230/8 = 57.5 \approx 58\$ gold.
Now you may wish to have more space than this, and perhaps dwarves in your world take higher rates or are unionized and only work 6 hour days. The general equation takes the form:
$$
\mathrm{cost = \frac{wage}{work day}\times\frac{\text{space per person}\times\text{height}\times\text{number of people}}{\text{excavation rate}}}
$$
You (or your DM) may wish to also account for smoothing the walls and so on, which might reasonably be accounted for as a doubling to the work amount (or halving the excavation rate).
You will also need to pay for furnishings and so on, which will be difficult to account for, and will likely depend of your standard. See the tables in chapter 5 of the Player's Handbook particularly adventuring gear and trade goods.
This should go without saying, but your DM has every right to set the cost higher than this. There are concerns not addressed here and the final say fall to your DM.
The closest I can find in the books to doing something like this is from Building a Stronghold in the DMG (p. 128) which puts a Town Hall or Trading post at 5,000 gp and 60 days. How exactly this translates into carving a house out of a cliff is difficult to say, but 2,000 gp and 30 days might be a good estimate for a decently sized home. (Cost and time does not scale linearly in the table.)
