How do you determine if a substrate will undergo SN1 reaction or E1 reaction??

Sn1 reactions are substituion reactions that have a unimolecular step as the rate determining step.  You can see that the rate determining step is the one that proceeds the slowest---it holds up the rest of the reaction from occuring.  It is a competitor with E1 as they both have the same unimolcular step.  Have I answered this clearly?

To determine which one will undergo Sn1 and E1 I think it has to do with the degree of substitution of the carbon, but I would have to look it up.

As far as differentiating, nucleophiles "attack the nucleus" of the carbocation....bases donate the proton for the reaction.  Typically step 3 of Sn1 reactions are acid base reactions

If there is a high degree of steric strain in the reactant then elimination will probably dominate. Also if elimination leads to a higher level of conjugation through the molecule (example 1-chloro-1,2-diphenylethane would probably undergo elimination as this leads to diphenylethene, indeed the dichloro compound can be used to prepare diphenylacetylene) The conjugate base of a strong acid Cl-, F- etc would tend to act as a nucleophile as abstraction of a proton woul lead to higher energy states (formation of a strong acid which in turn would want to protonate the alkene). Alkoxide would behave as both depending on the substrate. But to make a best guess you really just need experience.

Also if you wanted to promote elimination use a sterically hindered base as this limits the approach to the electrophilic carbon (lithium diisopropylamide is often used as this easily abstracts the very small proton but has essentially no interaction with, say a carbonyl group)

First of all SN1 has a 3 step mechanism while E1 has 2 step mechanism.  Look at solvent, SN1 has protic solvent (H is bonded to O or N), E1 is weak bases.  I think mostly look at the solvent and that E1 make a double bond or Pie bond while SN1 does not make a double bond.