Pleasure can be implicated in a chemical reaction in the brain, but does this really affect the intensity of the feeling? It is true that many, if not all feelings of our mind like euphoria, dysphoria, pleasure, sadness et cetera are linked to a chemical reaction in our brain. This reaction is essentially what causes us to feel these emotions. 

However, is the factor that causes us to really feel what we feel inherently biochemical? If so, then what neurobiological property of these reactions causes the intensity of the feeling? For example, what separates plain happiness from blissful ecstasy?
Short answer
In case of dopamine, its amounts present in the reward pathway in the brain correlates positively with the intensity of pleasure experienced.

Background
To focus the answer I'll narrow the scope of this post to one of the most extensively investigated emotional centers in the brain, namely the reward center involving the dopaminergic mesolimbic pathway.

Dopamine (DA) release in this system rewards the brain's host for beneficial things essential to life, such as eating, drinking and having sex. 

It is also one of the major neural centers in the brain involved in addiction, including nicotine, cocaine and methampetamine addiction, as well as gambling and pathological risk-taking behaviors. 

The opiate class of narcotics, most notably cocaine, amphetamine, methamphetamine, morphine, heroine and their analogues invoke a massive increase in DA levels in the mesolimbic pathway. An increase in DA levels has been shown to be a final (secondary) common pathway that mediates the rewarding properties of these drugs of abuse, explaining their addictive properties (Chiara &amp; Imperato, 1988).    

In humans it has been established by behavioral experiments that the magnitude of ventral striatal DA release (the striatum is a part of the reward system in the brain) correlates positively with the hedonic (pleasure) response associated with amphetamine (Drevets et al., 2011). In other words, more DA is correlated with more happiness and thus the amount of DA indeed correlates with the intensity of the feeling associated with its release in the reward system.  

As a side note on this; a notorious problem with many drugs of abuse are their purity. For example, Coca leaves cause a mild stimulant high when chewed. As far as I know, the American Indians have not suffered from overt physical or psychological dependence on these leaves in the thousands of years they probably have used them. However, shortly after the discovery of coca leaves (and opium poppies for that matter) by the Old World, the active ingredient was purified and eventually isolated. Cocaine (and morphine) cause a sudden and massive increase in dopamine levels (Romach et al., 1999), in turn not just leading to feelings of pleasure, but to near-instantaneous, full-blown ecstatic euphoria. It is this notorious 'first-time high' that virtually every addict keeps on pursuing, and which they will unfortunately never experience again in their life due to tolerance that build up almost instantly after first use of highly purified preparations of coca (crack, cocaine) and poppies (morphine, heroine).  

Post scriptum
In general, the 'biochemistry' of the brain refers to neurotransmitters. Their actions are primarily due the binding of these compounds to their respective receptors. The process of neurotransmitter release and receptor binding are not chemical reactions in the strictest sense of their meaning, as stated in your question. The secondary responses of the postsynaptic neuron may, or may not involve chemical reactions. Some second messenger systems do rely on chemical reactions (the metabotropic receptors including the dopamine receptors) while others do not (ion channel-coupled receptor systems for example). 

References
- Chiara &amp; Imperato, PNAS (1998); 85: 5274-8
- Drevets et al., Psychiatry (2001); 49: 81–96
- Romach et al., Arch Gen Psychiatry (1999); 56(12): 1101-6