Вопрос: Как из формы на PyQt5 ввести данные в PostgreSQL 9.4 в Debian 8 (только средствами PyQt5)?
Ответ: при помощи QSqlDatabase и QSqlQuery. Вот пример:

import sys
from PyQt5.QtWidgets import QDialog, QLineEdit, QPushButton, QFormLayout, QApplication
from PyQt5.QtSql import QSqlDatabase, QSqlQuery

class Form(QDialog):
    def __init__(self, parent=None):
        super(Form, self).__init__(parent)

        self.fn = QLineEdit()
        self.fn.setObjectName("firstname")
        self.fn.setText("John")

        self.ln = QLineEdit()
        self.ln.setObjectName("lastname")
        self.ln.setText("Smith")

        self.submit = QPushButton()
        self.submit.setObjectName("submit")
        self.submit.setText("Submit")

        layout = QFormLayout()
        layout.addWidget(self.fn)
        layout.addWidget(self.ln)
        layout.addWidget(self.submit)

        self.setLayout(layout)
        self.submit.clicked.connect(self.button_click)
        self.setWindowTitle("Add to PG")

    def button_click(self):
        fnstring = self.fn.text()
        lnstring = self.ln.text()
        request = "INSERT INTO table1 (fn,ln) VALUES('" + fnstring + "','" + lnstring + "');"
        print request

        db = QSqlDatabase.addDatabase("QPSQL")
        db.setHostName('localhost')
        db.setPort(5432)
        db.setDatabaseName('mydb')
        db.setUserName('mypguser')
        db.setPassword('mypgpassword')
        if db.isOpen():
            db.close()
        if not db.open():
            raise Exception("Error opening database: {0}".format(db.lastError().text()))
        query = QSqlQuery()
        query.exec_(request)

app = QApplication(sys.argv)
form = Form()
form.show()
app.exec_()