The naïve way of understanding mind uploading is "we take the connectome of a brain, including synaptic connection weights and characters, and emulate it in a computer". However, people want their personalities to be uploaded, not just brains. That is more than just replicating the functionality of their brains in silico.
This nuance has lead to some misunderstandings, for example, to experts wondering  why on Earth would anyone think that brain-centredness  (the idea that brains are "sufficient" in some vague sense) is a necessary prerequisite for successful whole brain emulation. Of course, brain-centredness is not required for brain uploading to be technically successful; the problem is whether it is sufficient for mind uploading in the sense that people actually care about?
The first obvious extension that may be required is the chemical environment of the brain. Here are some examples:
- Are you familiar with someone whose personality is radically (and often predictability) altered under influence of alcohol or drugs? This is not an exception, but a rule: most are impacted by this, only to a smaller extent. Only the transiency of the effects allow us to label them as simple mood changes.
- I have observed that my personal levels of neuroticism vary depending on the pharmaceutical drugs I'm using. Nootropics make me more nervous, while anti-hypertensions drugs have the reverse effect.
- The levels of hormones in the blood function as long-term personality changes. There are neurotransmitters that themselves are slow-acting, for example, nitric oxide .
- Artificially enchanted levels of serotonin in the brain causes it to "adapt" to this environment - in this way some of antidepressants work (namely, SSRI) .
Whole Brain Emulation - A Roadmap includes a short section about the "Body chemical environment" and concludes that for "WBE, the body chemistry model, while involved, would be relatively simple", unless protein interactions have to be modelled.
The technical aspect notwithstanding, what are the practical and moral implications? I think that here's not only a problem, but also an opportunity. Why keep the accidental chemistry we have developed in our lifetimes, one that presumably has little relation to what we would really like to be - if we could? Imagine that it is possible to create carefully improved and tailored versions of the neurotransmitter "soup" in the brain. There are new possibilities here for personal growth in ways that have not been possible before. These ways are completely orthogonal to the intelligence enhancement opportunities commonly associated with uploading.
The question of personal identity is more difficult, and there appears to be a grey zone here. A fictional example of the protagonist in Planescape: Torment comes into mind - is he the same person in each of his incarnations?
The second extension required to upload our personalities in the fullest sense might be the peripheral nervous system. Most of us think it's the brain that's responsible for emotions, but this is a simplified picture. Here are some hints why:
- The James-Lange 19th century theory of emotions proposed that we experience emotion in response to physiological changes in our body. For example, we feel sad because we cry rather than cry because we are sad . While the modern understanding of emotions is significantly different, these ideas have not completely gone away neither from academic research  nor everyday life. For example, to calm down, we are suggested to take deep and slow breaths. Paraplegics and quadriplegics, with severe spinal cord injuries typically experience less intense emotions than other people .
- Endoscopic thoracic sympathectomy (ETS) is a surgical procedure in which a portion of the sympathetic nerve trunk in the thoracic region is destroyed . It is typically used against excessive hand sweating. However, "a large study of psychiatric patients treated with this surgery [also] showed significant reductions in fear, alertness and arousal [..] A severe possible consequence of thoracic sympathectomy is corposcindosis (split-body syndrome) [..] In 2003 ETS was banned in Sweden due to overwhelming complaints by disabled patients." The complaints include having not been able to lead emotional life as fully as before the operation.
- The enteric nervous system in the stomach "governs the function of the gastrointestinal system" . I'm not sure how solid the research is, but there are a lot of articles on the Web that mention the importance of this system to our mood and well being . Serotonin is "the happiness neurotransmitter" and "in fact 95 percent of the body's serotonin is found in the bowels", as are 50% of dopamine . "Gut bacteria may influence thoughts and behaviour"  by using the serotonin mechanism. Also, "Irritable bowel syndrome is associated with psychiatric illness" .
In short, different chemistry in the brain changes what we are, as does the peripheral nervous system. To upload someone in the fullest sense, his/her chemistry and PNS also have to be uploaded.
 Anders Sandberg, Nick Bostrom, Future of Humanity Institute, Whole Brain Emulation - A Roadmap.
 Bradley Voytek's (Ph.D. neuroscience) Quora answer to Will human consciousness ever be transferrable?
 Selective serotonin reuptake inhibitors
 Bear et al. Neuroscience: Exploring the Brain, 3rd edition. Page 564.
 Michael W. Eysenck - Perspectives On Psychology - Page 100 - Google Books Result
 Endoscopic thoracic sympathectomy
 Enteric nervous system
 Scientific American, 2010. Think Twice: How the Gut's "Second Brain" Influences Mood and Well-Being
 The Guardian, 2012. Microbes manipulate your mind