How to sort character strings alphabetically?

for(int j=0;j<=size;j++)
{   
    cout<<"Mostrando persona:"<<j+1<<"\n";
    cout<<"Persona: "<<e[j].codigo<<"\n";
    cout<<"Persona: "<<e[j].nombre<<"\n";
    cout<<"Persona: "<<e[j].nombre_Materia<<"\n";
    cout<<"Persona: "<<e[j].nota_final<<"\n";
}

Reduce the number of iterations of this last loop by 1:

for(int j=0;j<size;j++)
//          ^^^^^^

On the other hand, notice that the ordering condition doesn't make much sense:

if( (e[i].nombre > e[j].nombre) || (e[i].nombre  == e[j].nombre && e[i].nombre> e[j].nombre) )

The ordering, I understand, is ascending, so this conditional can be simplified quite a bit, since you only have to check if the first name is bigger than the second:

if( (e[i].nombre > e[j].nombre) )

Otherwise the code seems to work fine.

I had a long time without using bubble so I took the code from this page: http://c.conclase.net/orden/?cap=burbuja Applying that code with your example, your order would be like this:

int i, j;
persona tempo;
for (int i = 0;i < size; i++){
    for (int j = 0; j< size-1; j++){
        if (e[j].nombre > e[j+1].nombre){
            tempo = e[j]; 
            e[j] = e[j+1]; 
            e[j+1] = tempo;
        }
    }
}

I have tried it and it has worked for me, as extra data, in your for where you ask the data you had <= and there it should only be the < because if it is not executed an extra time. I put the code of how I had the final code and apparently it worked, I hope it helps you

#include <iostream>
#include <string>

using namespace std;

struct persona
{

    int codigo;
    string nombre;
    string nombre_Materia;
    double nota1, nota2, nota3, nota_final;
};

persona e[7];

int main()
{

    int size;
    cout << "Ingrese el tamaño de Lista de estudiantes : ";
    cin >> size;
    for (int i = 0; i < size; i++) //tenias <= y se ejecutaba uno de mas
    {
        system("cls");
        cout << "Ingrese los datos de la persona: " << i + 1 << "\n";
        cout << "Ingrese el codigo: " << endl;
        cin >> e[i].codigo;
        cout << "Ingrese el nombre: " << endl;
        cin >> e[i].nombre;
        cout << "Ingrese el nombre de la materia: " << endl;
        cin >> e[i].nombre_Materia;
        cout << "Ingrese la nota 1: " << endl;
        cin >> e[i].nota1;
        cout << "Ingrese la nota 2: " << endl;
        cin >> e[i].nota2;
        cout << "Ingrese la nota 3: " << endl;
        cin >> e[i].nota3;
        e[i].nota_final = e[i].nota1 * 0.30 + e[i].nota2 * 0.30 + e[i].nota3 * 0.40;
    }

    int i, j;
    persona tempo;
    for (int i = 0;i < size; i++){
        for (int j = 0; j< size-1; j++){
            if (e[j].nombre > e[j+1].nombre){
                tempo = e[j]; 
                e[j] = e[j+1]; 
                e[j+1] = tempo;
            }
        }
    }

    for (int j = 0; j < size; j++) //tenias <= y se ejecutaba uno de mas
    {
        cout << "Mostrando persona:" << j + 1 << "\n";
        cout << "Persona: " << e[j].codigo << "\n";
        cout << "Persona: " << e[j].nombre << "\n";
        cout << "Persona: " << e[j].nombre_Materia << "\n";
        cout << "Persona: " << e[j].nota_final << "\n";
    }
};

Your approach to the problem is more complicated than necessary. Simply define the ordering relationship between instances of personaand delegate the ordering to a standard library algorithm .

Let's start by defining a less than operator ( <) for objects persona:

bool operator <(const persona &a, const persona &b) {
    return a.nombre.compare(b.nombre) < 0;
}

With this operator defined, you will be able to sort:

std::sort(std::begin(e), std::end(e));

Your code might look something like this:

struct persona {
    int codigo;
    std::string nombre;
    std::string nombre_Materia;
    double nota1,nota2,nota3,nota_final;
};

bool operator <(const persona &a, const persona &b) {
    return a.nombre.compare(b.nombre) < 0;
}

int main()
{
    persona e[7]{};

    for (auto &p : e)
    {
        std::cin >> p.codigo;
        std::cin.ignore(1);
        std::getline(std::cin, p.nombre);
        std::getline(std::cin, p.nombre_Materia);
        std::cin >> p.nota1;
        std::cin >> p.nota2;
        std::cin >> p.nota3;
        std::cin >> p.nota_final;
    }

    std::sort(std::begin(e), std::end(e));

    for (const auto &p : e)
        std::cout << p.codigo << ' ' << p.nombre << '\n'
            << p.nombre_Materia << ": " << p.nota1 << ' ' << p.nota2 << ' ' << p.nota3 << ' ' << p.nota_final << '\n';

    return 0;
}

You can see it working in TIO.