That's probably because the paint bucket tool is filling the partial transparent pixels on the edges.
If that is on a separate layer you could lock the transparent pixels before filling or use the fill path command again with a different color.
Probably the best solution is to convert your path to a shape layer (color fill).
The resolution of the file will have an impact if you fill with the paint bucket.
Do target the path, and add a color fill layer (Layer>New fill layer>Solid color...
R_Kelley is correct in that the root of your problem is transparent pixels being filled.
There are two very fast and easy ways to change the fill of an object without using the paint bucket tool.
1) The first, as PECourtejoie recommended, is to add a fill layer. Layer > New Fill Layer > Solid Color... should bring up a dialogue box like the one below. Just make sure to have your shape layer selected before you do this, and check the "Use Previous Layer to Create Clipping Mask" option.
This option allows you to go back and change the color of the fill layer by double clicking on the clipping mask icon (the left box in the screenshot above).
2) The second option is to add a color overlay Layer Style to your object. Just go to Layer > Layer Style > Color Overlay... This color overlay can be changed later by opening up the same dialogue box.