This is a somewhat unorthodox approach (and unlist(strsplit(...))
would be very fast anyway), but you can use one of the string searching packages that offer vectorized search pattern options, like "stringi":
## Read the data in. Since it's not a data.frame, just use readLines
X <- readLines("http://users.utu.fi/jjahol/chromosome.txt")
## Paste the lines together into a single block of text
Y <- paste(X, collapse = "")
library(stringi)
Strings <- c("A", "C", "G", "T")
stri_count_fixed(Y, Strings)
# [1] 15520 13843 14215 16422
## Named output....
setNames(stri_count_fixed(Y, Strings), Strings)
# A C G T
# 15520 13843 14215 16422