Darwin did this by translating a hierarchy of (anatomical) similarity into a tree scheme that linked organisms from successive time slices with thousands of generations separating each two slices (see the scheme from Darwin’s "Origin of Species": [link]). The time slices can be related to certain units of the geological time scale.
Deep time correspondents in stratigraphy …The problem of imagining time ranges far outside the scale of human experience has been approached by geologists with the method and concepts of stratigraphy: Strata of rock can be interpreted as a succession of time slices. Relative ages and age differences often manifest in an amount of rock which is loosely corresponding to certain a time span if similar rockforming processes are underlying. The relationship between the duration of a process and the amount of materal it creates can be inferred from direct observation of recent systems, allowing the assignment of absolute time (in years or millions of years) to a succession of strata. Given that long-term geological processes are rarely gradual, a more reliable absolute age is provided by radiometric dating.
If you would ask a geologist how he/ she percieves deep time I suppose he/ she would explain that it becomes clear from the slowness of present-day geological processes on the one hand and from the vast amount of products of such processes on the other hand.
… and phylogeneticsThe evolution of organisms yields another approximation of deep time: The passing of time manifests in the hierarchical distinctness of living systems. Seeing how slow evolution works in a human being’s life span and how much change in anatomy/ biochemistry etc. must have occurred since last common ancestor of mouse and elephant or of mouse and lemon tree, leads to another way of percieving long time spans.
Regarding a certain distinctness and species richness of a group as a product of a certain number of character changes and speciation events (which is more or less well correlated with time) was probably enhanced by the more quantitative look at phylogenetics since the introduction of cladistics and molecular methods.
When I was attending a workshop on molecular paleobiology in 2008 specialists of that field were using the expression “(addressing) deep time problems” synonymous to phylogenetics of higher systematic groups, i.e. as the study of evolutionary changes that occurred deep down in the tree of animals and other organisms – opposed to let’s say the comparative analysis of human and neanderthal genomes or the radiation of Darwin finches.
The viewpoints of paleontologists…But how do (present-day) paleontologists percieve deep time? You would expect them to share the view of both, phylogeneticists and stratigraphers, as most of them are taught at least a bit about both fields. However, for a paleontological fieldworker who employs the study of fossils as a means to understand and describe geological processes and paleoenvironmental contexts the flow of time is much easier grasped as a series of events preserved in a succession of rocks (and not as a phylogenetic tree scheme).
On the other hand, as a consequence of the so-called paleobiological revolution, you don’t need to be a field worker to contribute to the understanding of ancient organisms. In fact many aspects of paleobiology require mere laboratory and magazine work and you can spend a lifetime on that without ever considering rocks – naturally the perspective of such a modern paleobiologist on deep time will be strictly that of a phylogeneticist.
…can lead to conflicts?These different perceptions on deep time and evolution are probably the background why cladistics was (and still is) met with some scepticism by “old school paleontologists” (or by “Eastern Europe school paleontologists”): Instead of considering all kinds of data for phylogenetic hypothesis-making I am supposed to use merely data from the (anatomical, molecular, etc.) comparison of organisms, as if evolution does not manifest in other ways in the geological record.
One could argue that it is possible to integrate other data, i.e. stratigraphic ages and palaeobiogeographical relations, in a cladistic analysis or at least in the discussion of its results, and so assure that hypotheses from the tree perspective on evolution are tested under consideration of independent data.
The idea to use time directly as a character in a parsimony analysis with consecutive time slices as character states may be epistemically unsound, as it is problematic to justify any kind of model assumption how time is weighted with respect to anatomical characters (and implicitly would this mean a post-hoc failure if proximity in time is regarded as indicative for degree of relationship?).
An a posteriori fit to other data – e.g. looking which of the equally parsimonious morphology-only-based time-calibrated trees has shorter lineages of no record (ghost lineages) – might be a better approach, but is still hard to swallow for some people who have problems with parsimony analyses on the basis of (too) small character samples (i.e. with inherent biases due to sample size/ character poorness or ambivalence of fossils).