If I understood the question correctly then 
For hourly frequency:
library(dplyr)
df %>%
  mutate(start_timestamp = as.POSIXct(paste(df$StartDate, df$StartTime), tz="UTC", format="%Y-%m-%d %H")) %>%
  right_join(data.frame(seq_h = as.POSIXct(unlist(lapply(unique(df$StartDate), 
                                                         function(x) seq(from=as.POSIXct(paste(x, "00:00:00"), tz="UTC"),
                                                                         to=as.POSIXct(paste(x, "23:00:00"), tz="UTC"),
                                                                         by="hour"))), origin="1970-01-01", tz="UTC")), by=c("start_timestamp" = "seq_h")) %>%
  group_by(start_timestamp) %>%
  summarise(freq=sum(!is.na(TripId)))
Output is:
   start_timestamp      freq
 1 2016-01-01 00:00:00     2
 2 2016-01-01 01:00:00     1
 3 2016-01-01 02:00:00     1
 4 2016-01-01 03:00:00     0
 5 2016-01-01 04:00:00     0
...
For two-hourly frequency:
library(dplyr)
df %>%
  mutate(start_timestamp = as.POSIXct(cut(as.POSIXct(paste(df$StartDate, df$StartTime), tz="UTC"), breaks="2 hours"), tz="UTC")) %>%
  right_join(data.frame(seq_h = as.POSIXct(unlist(lapply(unique(df$StartDate), 
                                                         function(x) seq(from=as.POSIXct(paste(x, "00:00:00"), tz="UTC"),
                                                                         to=as.POSIXct(paste(x, "23:00:00"), tz="UTC"),
                                                                         by="2 hours"))), origin="1970-01-01", tz="UTC")), by=c("start_timestamp" = "seq_h")) %>%
  group_by(start_timestamp) %>%
  summarise(freq=sum(!is.na(TripId)))
Output is:
   start_timestamp      freq
 1 2016-01-01 00:00:00     3
 2 2016-01-01 02:00:00     1
 3 2016-01-01 04:00:00     0
 4 2016-01-01 06:00:00     0
 5 2016-01-01 08:00:00     0
...
Sample data:
df <- structure(list(TripId = c(15335543L, 15335544L, 15335607L, 15335608L, 
15335613L, 15335639L), StartDate = c("2016-01-01", "2016-01-01", 
"2016-01-01", "2016-01-01", "2016-01-02", "2016-01-02"), StartTime = c("00:14:00", 
"00:14:00", "01:00:00", "02:01:00", "02:16:00", "02:50:00")), .Names = c("TripId", 
"StartDate", "StartTime"), class = "data.frame", row.names = c("1", 
"2", "3", "4", "5", "6"))