How to combine ggplot and dplyr into a function?

ggplot does not yet support tidy eval syntax (you can't use the !!). You need to use more traditional standard evaluation calls. You can use aes_q in ggplot to help with this.

get_charts1 <- function(data, mygroup){

  quo_var <- enquo(mygroup)

  df_agg <- data %>% 
    group_by(!!quo_var) %>% 
    summarize(mean = mean(value, na.rm = TRUE),
              count = n()) %>% 
    ungroup()

  ggplot(df_agg, aes_q(x = quote(count), y = quote(mean), color = quo_var, group = quo_var)) + 
    geom_point() +
    geom_line() 
}


get_charts1(dataframe, group)

ggplot2 v3.0.0 released in July 2018 supports !! (bang bang), !!!, and :=. aes_()/aes_q() and aes_string() are soft-deprecated.

OP's original code should work

library(tidyverse)

get_charts1 <- function(data, mygroup){

  quo_var <- enquo(mygroup)

  df_agg <- data %>% 
    group_by(!!quo_var) %>% 
    summarize(mean = mean(value, na.rm = TRUE),
              count = n()) %>% 
    ungroup()

  ggplot(df_agg, aes(x = count, y = mean, 
                color = !!quo_var, group = !!quo_var)) + 
    geom_point() +
    geom_line() 
}

get_charts1(dataframe, group)

Edit: using the tidy evaluation pronoun .data[] to slice the chosen variable from the data frame also works

get_charts2 <- function(data, mygroup){

  df_agg <- data %>% 
    group_by(.data[[mygroup]]) %>% 
    summarize(mean = mean(value, na.rm = TRUE),
              count = n()) %>% 
    ungroup()

  ggplot(df_agg, aes(x = count, y = mean, 
                     color = .data[[mygroup]], group = .data[[mygroup]])) + 
    geom_point() +
    geom_line() 
}

get_charts2(dataframe, "group")

^{Created on 2018-04-04 by the reprex package (v0.2.0).}