Why are negatively charged ligands behind the neutral ligands in spectrochemical series?
I think you are confused by what the spectrochemical series actually shows. It is not a rating of stronger or less strong ligands to mean they can bind more or less strongly to the metal. It is only a rating of which ligands induce a high crystal field split in the resulting complex.

The splitting derives from the metal $\mathrm{e_g}$ orbitals — $\mathrm{d}_{x^2 - y^2}$ and $\mathrm{d}_{z^2}$ — actually being antibonding to the metal-ligand interaction and thus higher in energy than the (unmodified in a complex with only &#x3c3; type interactions) $\mathrm{t_{2g}}$ orbitals.

For systems more complex, i.e. including &#x3c0; interactions, the $\mathrm{t_{2g}}$ orbitals (the other three d-orbitals) interact in a &#x3c0; manner with ligand p-orbitals. If the ligand has populated p-orbitals such as halides have, the $\mathrm{t_{2g}}$ energy is raised as it is turned into a slightly antibonding interaction (the bonding one being on the ligands’ side). Since this brings $\mathrm{t_{2g}}$ closer to $\mathrm{e_g}$, the field split is reduced and the ligands are termed weak-field ligands. In simplified Lewis schemes, one could represent this by a double forwards bond from the ligands to the metal.

In the case of empty p-type orbitals such as in $\ce{CO}$, metal to ligand back-bonding can occur by populated metal orbitals interacting with unfilled ligand orbitals; now it is the metal’s side of things that is stabilised. Since this &#x3c0; backbonding stabilises the $\mathrm{t_{2g}}$ orbitals, their energy is lowered and the field split becomes greater. Hence they are termed strong-field ligands. In simplified Lewis structures one could represent that by drawing a donor-acceptor arrow from the ligand to the metal (&#x3c3; forwards bond) and one the other way (&#x3c0; back bond).

This all has nothing to do with the overall strength of metal-ligand interaction, as I mentioned earlier. In fact, strong complexes are typically obtained for certain metal-ligand combinations and the same ligand can be very bad in interacting with other metals. Classic example: $\ce{CO}$ ligands are extremely good at stabilising low metal oxidation states such as in $\ce{[Fe(CO)4]^2-}$, while they are rather lousy in stabilising high ones: $\ce{[Fe(CO)6]^2+}$ can only be synthesised in the extreme absence of all other ligands, i.e. in $\ce{SbF5}$ from $\ce{FeF2}$ where the fluorides will be drawn towards antimony to generate the even more stable $\ce{[SbF6]-}$ complex.