Consistency of consciouness
You trespass upon a question as old as antiquity: what properties constitute consciousness? The best means to respond to the question concerns a metaphysical answer and an answer in modern biology

We can perhaps divide consciousness into two categories: contiguous thought from past experience and identity and secondly, self-awareness.  

The reputed example of the Sorites problem of Theseus' Ship describes how the physical entity of Theseus' ship, during his various battles and broils, loses certain components of his ship: his braces or mast, in example. They are replaced with exact replicates until the time that every component of the ship has been supplanted by a replaced component. Is it the same ship? It is dependent on your definition of identity. 

Leibniz's Law of Identity whether in humans, ships or spaghetti monsters, states that objects x and y hold the same identity if and only if they share all the same predicates - qualities or attributes. If this be of functionality, the ship completes the same function and comprises the same components, albeit of different origin - thus, the ship has the same identity. This is important regarding your question whether your 'old [you] dies'. Over time, your cells undergo mitosis and eventually apoptosis. Hence, although your parts are in constant flux, the predicates between your old self and latter self are constant. Ideologically however, there is no set identity! 

If you read David Hume's literature, you will observe the notion that one is only the composition of impressions (sensory reception) and thus there exists no identity.   

Your question regarding clones is a challenge for biology. Behaviourism is an approach in biology concerning the interpretation of behaviour as a response to a stimulus. Furthermore, humans express complex instinctive behaviours in response to various stimuli. If there exists no 'mental substance' distinct from the body, then it seems plausible to consider human consciousness merely an extension of an advanced instinctive epiphenomenon: self-awareness is a by-product to these advanced behavioural patterns. To support this position, simply recall what factors allow one to discriminate individuals whom one meets, behavioural patterns and appearance.  

Finally, one could attempt to assess consciousness in regard to self-awareness by conducting the mirror assessment. If one is capable of identifying himself/herself, then it is indicative of an ability to discern one's personal existence from general matter.

This is by no means a comprehensive answer to this question yet I bid you well in your continued explorations of human consciousness.