You can influence the appearance of the titles in the table of contents with the titletoc package. This allows you to remove the number.
The command for this is \titlecontents. See the comments in the code below for a description of the arguments of this command. See also the package documentation.
MWE (note that I removed the manual 'Chapter n.' parts of the \chapter commands):
\documentclass[a4paper, 12pt, oneside, openany]{book}
\usepackage[a4paper, inner=4cm, outer=3cm, top=4cm, bottom=3cm]{geometry}
\usepackage{blindtext}
\usepackage{titletoc}
\titlecontents{chapter}% redefine chapter toc appearance
[1.5em]% left margin
{}% code before starting the title, here empty
{\hspace*{-2.3em}\bfseries}% format: shift a bit left and boldface, no number
{\hspace*{-2.3em}\bfseries}% format for \chapter*, not really needed for the MWE but added for completeness
{\titlerule*[1pc]{.}\contentspage}% dotted line and page number
\begin{document}
\tableofcontents
\chapter{INTRODUCTION}
\section{Section Name}
\blindtext
\section{Section Name}
\blindtext
\chapter{BACKGROUND}
\section{Section Name}
\blindtext
\section{Section Name}
\blindtext
\end{document}
Result in TOC:

In the document:

If you also want to change the appearance in the document then you can use the titlesec package which is closely related to titletoc with very similar syntax. The command for changing the title in the document is \titleformat. See the comments in the code below for a description of the arguments of this command.
\documentclass[a4paper, 12pt, oneside, openany]{book}
\usepackage[a4paper, inner=4cm, outer=3cm, top=4cm, bottom=3cm]{geometry}
\usepackage{blindtext}
\usepackage{titlesec}
\usepackage{titletoc}
\titlecontents{chapter}% redefine chapter toc appearance
[1.5em]% left margin
{}% code before starting the title, here empty
{\hspace*{-2.3em}\bfseries}% format: shift a bit left and boldface, no number
{\hspace*{-2.3em}\bfseries}% format for \chapter*, not really needed for the MWE but added for completeness
{\titlerule*[1pc]{.}\contentspage}% dotted line and page number
\titleformat{\chapter}% change format of chapter in the document
{\huge\bfseries}% huge font and bold
{Chapter \thechapter}% the word Chapter followed by the chapter number
{1em}%horizontal space between 'Chapter n' and the chapter title
{}% code after the title, here empty
\begin{document}
\tableofcontents
\chapter{INTRODUCTION}
\section{Section Name}
\blindtext
\section{Section Name}
\blindtext
\chapter{BACKGROUND}
\section{Section Name}
\blindtext
\section{Section Name}
\blindtext
\end{document}
The TOC is the same, but in the document a chapter is now as follows:

}afterbook– daleif Jul 31 '20 at 09:40Chapter 1. INTRODUCTIONas well. (No number behind the title). – user516076 Jul 31 '20 at 09:43Chapter 1,Chapter 2by hand? That makes no sense. – daleif Jul 31 '20 at 10:08\chapter*{...}because it makes the first section start with0.1, 0.2 ....instead of1.1, 1.2, 1.3. And an easy way to get1.1, 1.2, 1.3is using\chapter{...}and writing chapter by hand. – user516076 Jul 31 '20 at 10:16\section*{...}to avoid the zero starts. But that's a bad idea cuz it's not automatic and another reason. I just want to remove that number behind the title of chapter, that's all. Please help me. – user516076 Jul 31 '20 at 10:22\refstepcounter{chapter}or\stepcounter{chapter}– daleif Jul 31 '20 at 10:27\stepcounter{chapter}and it doesn't work for chapter 2 – user516076 Jul 31 '20 at 10:40\refstepcounter{chapter}or\stepcounter{chapter}, the section only shows1.1, 1.2....and can't show2.1, 2.2. That's why i'm using\chapter{...}– user516076 Jul 31 '20 at 10:49\chapter{....}andsection{...}. I managed to fix the text ouput on the mainmatter, so all i care about is only to remove the number behind the chapter. – user516076 Jul 31 '20 at 10:57