Will a chin implant make my head look larger? I have a weak chin that I want to fix but I also have a cone shaped head. (photos)

A chin implant will improve projection and overall shape of the chin, but won't improve the shape of your head. You have a few options to improve the projection and symmetry of your chin and profile. If you’re not sure you’re ready for surgery, but still want to improve and strengthen your chin, I’d suggest augmenting your chin with Sculptra and/or Voluma. If you like your new look, you can make it permanent with a chin implant. I hope that helps.

Whether you interpret if a chin implant would make your head look larger can be determined before surgery by computer imaging. While there is no direct correlation between the chin and the back of the head per se, how you may see it is what is important. Also be aware that the back of your head projection can also be reduced at the time of a chin implant placement if its placement is perceived as being aesthetically detrimental to the head shape.
FYI putting on weight will not change the thickness of your neck.

 Hi, I have performed many facial shaping procedures using dermal fillers, facial implants (cheek, chin), liposuction and/or facelifts for over 30 years.  Following my beauty principles, men look chiseled and handsome with angularity in the cheeks, chin and mandibular angles.  From the photos, your cheeks are concave in the front, the chin is quite weak as are the mandibular angles.  These would typically combine to create an elongated oval or rectangualrly-shaped face which can be confirmed with a non smiling photo of your face from the front.  The following procedures would create a more ruggedly handsome face.
Cheek augmentation with a dermal filler or silastic cheek implants to add angularity to the cheeks and mid face.  
Chin augmentation, using a chin implant, to add projection to the chin creating harmony and balance to the lower face.  I have performed many Chin Augmentations with dermal fillers or silastic chin implants, for over 30 years.  When the chin is weak, this creates an imbalance making the nose appear larger, the mid face top heavy, the lower face looks short, de-emphasizes the lips and allows early formation of a "double chin".  Proper placement of a silastic chin implant adds forward projection to the chin thereby creating harmony and balance to the lower face.  Using the same incision, liposuction can be performed to reduce the fat and further shape the neck.  Excess skin, from below the chin, can also be removed through the same incision.  I have found that placement of a silastic chin implant, through a small curved incision under the chin (also allows excess skin removal) to be very safe, quick, highly effective and far less invasive than a sliding genioplasty (requires extensive tissue dissection, bone cuts and placement of metal screws and plates to secure the cut bone segments).  I perform chin implant surgery in 30 minutes or less, often using a local anesthetic alone.  In my opinion, you are a good candidate for chin implant surgery.  Proper chin implant placement causes a "forward" projection that balances the lower face.  It would not increase the "elongated" appearance of the face...in fact it helps to correct it.
Augmentation of the mandibular angles using a dermal filler (jaw implants have too many unwanted side effects IMHO) to add volume, angularity and flare to the jaw line.There's nothing that can or should be done to the occipital skull.
Hope this helps.