First, on handling the lines separately:
Example:
if (myfile.is_open())
{
    short line_num = 1;
    for (std::string line; std::getline(myfile, line); ) {
        cout << "Line number: " << line_num << " Value: " << line << std::endl;
        line_num++;
    }
}
Output:
mindaugasb@c_cpp:~/workspace/StackOverflow/ReadingAFileLineByLine $ ./r2v.exe 
Please enter file name!
f
Line number: 1 Value: 123, 123, 431, 5123, 12312
Line number: 2 Value: 25316, 64234, 1231, 124123
Second on pushing the lines, item at a time, to a vector:
I will assume there will be two vectors, for the two lines in the input file. If there are more lines, you will need to create more vectors, possibly by implementing a function that will accept as string (obtained from the input file) and return a vector in the form you want it. Example:
vector<string> line2Vec(string line)
{
    char * tokens;
    char * line_chararr = const_cast< char *>(line.c_str());   
    tokens = strtok(line_chararr, ", ");
    vector<string> vs;
    while (tokens != NULL)
    {
        vs.push_back(tokens);
        tokens = strtok (NULL, ", ");
    }
    return vs;
}
Attention - I do not say that this is perfect Cpp code or a perfect solution to your problem. However it is A solution to how I understood the problem based on what was written. Here is the full version to play around with:
#include <iostream>
#include <fstream>
#include <stdlib.h>
#include <sstream>
#include <vector>
#include <string.h>
using namespace std;
vector<string> line2Vec(string line)
{
    cout << line << endl;
    char * tokens;
    char * line_chararr = const_cast< char *>(line.c_str());   
    tokens = strtok(line_chararr, ", ");
    vector<string> vs;
    while (tokens != NULL)
    {
        vs.push_back(tokens);
        tokens = strtok (NULL, ", ");
    }
    return vs;
}
int main()
{
    string fname;
    cout << "Please enter file name!" << endl;
    cin >> fname;
    fstream  myfile;
    myfile.open(fname);
    if (myfile.is_open())
    {
        for (std::string line; std::getline(myfile, line); ) {
            line2Vec(line);
        }
    }
    else
    {
        cout << "Unable to open your file!" << endl;
    }
}