Unfortunately nothing in the bonding situation in carbon monoxide is easily explained, especially not the dipole moment. According to the electronegativities of the elements you would expect the partial positive charge to be at the carbon and a partial negative charge at oxygen. However, this is not the case, which can only be explained by molecular orbital theory. A complete analysis of this can be found in Gernot Frenking, Christoph Loschen, Andreas Krapp, Stefan Fau, and Steven H. Strauss, J. Comp. Chem., 2007, 28 (1), 117-126. (I believe it is available free of charge.) Responsible for the dipole moment is the highest occupied molecular orbital, a $\pmb{\sigma}$ orbital, which has its largest coefficient at the carbon atom. In first order approximation, this orbital can be considered the lone pair of carbon. All other valence orbitals are more strongly polarised towards the oxygen. The orbital that can in first order approximation be considered as the oxygen lone pair has almost only s character and therefore contributes only little to the dipole moment. \begin{align} \ce{{}^{\ominus}\!:C#O:^{\oplus}} && \text{Dipole:}~|\mathbf{q}|=0.11~\mathrm{D} && \text{Direction:}~\longleftarrow \end{align}

I have reproduced the MO scheme of carbon monoxide for you below. Please note, that the blue/orange coloured orbitals are virtual (unoccupied) orbitals, which should be taken with a grain of salt.

There are two possible decomposition schemes to explain the bonding, both of them involve donor-acceptor interactions. The term "dative bonding" should be avoided here, it is better to use it only for bonds, that consist purely of donor-acceptor interactions, as for example in $\ce{H3N\bond{->}BH3}$.

Below the two decomposition schemes are reproduced from figure 6 (b & c) in the linked paper. Please note, that this decomposition does not include hybridised orbitals.

The left decomposition is a better description, since it retains the $C_{\infty{}v}$ symmetry if the molecule. We can see a donor-acceptor $\sigma$ bond and two electron sharing $\pi$ bonds.
In the right configuration we assume an electron sharing $\sigma$ bond, an electron sharing $\pi$ bond and a donor-acceptor $\pi$ bond.

It is very important to understand, that the concept of a dative bond, that you are trying to employ here is only right by coincidence. The reason that the dipole moment is oriented towards the carbon is only to find in the weakly bonding HOMO, the lone pair of carbon.

Unfortunately, nothing in the bonding situation in carbon monoxide is easily explained, especially not the dipole moment. According to the electronegativities of the elements, you would expect the partial positive charge to be at the carbon and a partial negative charge at oxygen. However, this is not the case, which can only be explained by molecular orbital theory. A complete analysis of this can be found in Gernot Frenking, Christoph Loschen, Andreas Krapp, Stefan Fau, and Steven H. Strauss, J. Comp. Chem., 2007, 28 (1), 117-126. (I believe it is available free of charge.) Responsible for the dipole moment is the highest occupied molecular orbital, a $\pmb{\sigma}$ orbital, which has its largest coefficient at the carbon atom. In first order approximation, this orbital can be considered the lone pair of carbon. All other valence orbitals are more strongly polarised towards the oxygen. The orbital that can in first order approximation be considered as the oxygen lone pair has almost only s character and therefore contributes only little to the dipole moment. \begin{align} \ce{{}^{\ominus}\!:C#O:^{\oplus}} && \text{Dipole:}~|\mathbf{q}|=0.11~\mathrm{D} && \text{Direction:}~\longleftarrow \end{align}

I have reproduced the MO scheme of carbon monoxide for you below. Please note, that the blue/orange coloured orbitals are virtual (unoccupied) orbitals, which should be taken with a grain of salt.

There are two possible decomposition schemes to explain the bonding, both of them involve donor-acceptor interactions. The term "dative bonding" should be avoided here, it is better to use it only for bonds, that consist purely of donor-acceptor interactions, as for example in $\ce{H3N\bond{->}BH3}$.

Below, the two decomposition schemes are reproduced from figure 6 (b & c) in the linked paper. Please note, that this decomposition does not include hybridised orbitals.

The left decomposition is a better description, since it retains the $C_{\infty{}v}$ symmetry of the molecule. We can see a donor-acceptor $\sigma$ bond and two electrons sharing $\pi$ bonds.
In the right configuration we assume an electron sharing $\sigma$ bond, an electron sharing $\pi$ bond and a donor-acceptor $\pi$ bond.

It is very important to understand, that the concept of a dative bond, that you are trying to employ here is only right by coincidence. The reason that the dipole moment is oriented towards the carbon is only to find in the weakly bonding HOMO, the lone pair of carbon.