set.seed(100)
df <- data.frame(cat = c(rep("aaa", 5), rep("bbb", 5), rep("ccc", 5)), val = runif(15))
df <- df[order(df$cat, df$val), ]
df

   cat        val
1  aaa 0.05638315
2  aaa 0.25767250
3  aaa 0.30776611
4  aaa 0.46854928
5  aaa 0.55232243
6  bbb 0.17026205
7  bbb 0.37032054
8  bbb 0.48377074
9  bbb 0.54655860
10 bbb 0.81240262
11 ccc 0.28035384
12 ccc 0.39848790
13 ccc 0.62499648
14 ccc 0.76255108
15 ccc 0.88216552

df$num <- 1
 for (i in 2:(length(df[,1]))) {
   if (df[i,"cat"]==df[(i-1),"cat"]) {
     df[i,"num"]<-df[i-1,"num"]+1
     }
 }
 df

   cat        val num
1  aaa 0.05638315   1
2  aaa 0.25767250   2
3  aaa 0.30776611   3
4  aaa 0.46854928   4
5  aaa 0.55232243   5
6  bbb 0.17026205   1
7  bbb 0.37032054   2
8  bbb 0.48377074   3
9  bbb 0.54655860   4
10 bbb 0.81240262   5
11 ccc 0.28035384   1
12 ccc 0.39848790   2
13 ccc 0.62499648   3
14 ccc 0.76255108   4
15 ccc 0.88216552   5

df$num <- ave(df$val, df$cat, FUN = seq_along)

library(plyr)
ddply(df, .(cat), mutate, id = seq_along(val))

library(dplyr)
df %>% group_by(cat) %>% mutate(id = row_number())

library(data.table)
DT <- data.table(df)

DT[, id := seq_len(.N), by = cat]
DT[, id := rowid(cat)]

df$num <- sequence(rle(df$cat)$lengths)

> df
   cat        val num
4  aaa 0.05638315   1
2  aaa 0.25767250   2
1  aaa 0.30776611   3
5  aaa 0.46854928   4
3  aaa 0.55232243   5
10 bbb 0.17026205   1
8  bbb 0.37032054   2
6  bbb 0.48377074   3
9  bbb 0.54655860   4
7  bbb 0.81240262   5
13 ccc 0.28035384   1
14 ccc 0.39848790   2
11 ccc 0.62499648   3
15 ccc 0.76255108   4
12 ccc 0.88216552   5

df$num <- sequence(rle(as.character(df$cat))$lengths)

for (i in unique(df$cat)) df$num[df$cat == i] <- seq_len(sum(df$cat == i))

# 1. Data set
set.seed(100)
df <- data.frame(
  cat = c(rep("aaa", 5), rep("ccc", 5), rep("bbb", 5)),
  val = runif(15))

# 2. 'dplyr' approach
df %>%
  arrange(cat, val) %>%
  group_by(cat) %>%
  mutate(id = row_number())

# Variant with ascending ordering
library(data.table)
dt <- data.table(df)
dt[, .( val
   , num = rank(val))
    , by = list(cat)][order(cat, num),]

    cat        val num
 1: aaa 0.05638315   1
 2: aaa 0.25767250   2
 3: aaa 0.30776611   3
 4: aaa 0.46854928   4
 5: aaa 0.55232243   5
 6: bbb 0.17026205   1
 7: bbb 0.37032054   2
 8: bbb 0.48377074   3
 9: bbb 0.54655860   4
10: bbb 0.81240262   5
11: ccc 0.28035384   1
12: ccc 0.39848790   2
13: ccc 0.62499648   3
14: ccc 0.76255108   4

# Variant with descending ordering
dt[, .( val
   , num = rank(-val))
    , by = list(cat)][order(cat, num),]