In a secp256k1 context (including bitcoin), multiplying a point $P$ on the curve by an integer $k$ leads to the point at infinity $\mathcal O$ if and only if:
- $k$ is a multiple of the group order $n$, including when $k=0$. This $n$ is a large (prime) integer part of the secp256k1 characteristics. That's the number of points on the curve, including the point at infinity.
- or point $P$ is the point at infinity $\mathcal O$. That can't occur when $P$ is the generator $G$.
Mathematically: $k\times P=\mathcal O\iff k\bmod n=0$ or $P=\mathcal O$.
A good analogy is multiplication modulo prime $n$: multiplication by $k$ yields zero (modulo $n$) when $k$ is a multiple of $n$, or when what we multiply was already zero (modulo $n$). The analogy goes further: adding something that's zero (modulo $n$) changes nothing (modulo $n$), and that's the definition of zero (modulo $n$); much like adding the point at infinity $\mathcal O$ to any point on the curve leaves it unchanged, and that's the definition of the point at infinity.
Even simpler analogy: when in clock arithmetic we repeatedly add the current time (expressed as a whole number of hours), or equivalently multiply the current time by increasingly large integers, we eventually get to 12 o'clock (even if we did not start from 12 o'clock, that can occur before we multiplied by 12; that's because 12 is not prime).
Notice that when multiplying by $k$ not a multiple of $n$, some point multiplication algorithms might still internally encounter the point at infinity. For example when $k=8n-1$, some algorithms to compute $k\times P$ could compute it as $2\times\bigl(2\times\bigl(2\times\bigl(((k+1)/8)\times P\bigr)\bigr)\bigr)\,-\,P$. This is mathematically correct, but encounters the point at infinity in the computation of $((k+1)/8)\times P$.
When $P$ is not on the curve, multiplying it by an integer is not mathematically defined, and what happens can only we stated by examining how the multiplication is attempted.
Some point multiplication algorithms avoid both issues by special-casing multiplying the point at infinity; validating that the point is on the curve; reducing $k$ modulo $n$, then special-casing $k=0$; and making sure that they internally manipulate $j\times P$ only with $|j|<n$ (perhaps with an exception for even $j$ with $|j|<2n$ ).