Building Dynamic Oracle Where Clause

This might give you an idea:

create table Customer (
  c_firstname varchar2(50),
  c_lastname  varchar2(50),
  c_userid    varchar2(50)
);

insert into Customer values ('Micky' , 'Mouse', 'mm');
insert into Customer values ('Donald', 'Duck' , 'dd');
insert into Customer values ('Peter' , 'Pan'  , 'pp');

create or replace function GetCustomer(
  FirstN    varchar2 := null,
  LastN     varchar2 := null,
  CID       varchar2 := null
) return sys_refcursor
as
  stmt varchar2(4000);
  ret sys_refcursor;
begin
  stmt := 'select * from Customer where 1=1';

  if  FirstN is not null then
      stmt := stmt || ' and c_firstname like ''%' || FirstN || '%''';
  end if;

  if  LastN is not null then
      stmt := stmt || ' and c_lastname like ''%' || LastN  || '%''';
  end if;

  if  CID is not null then
      stmt := stmt || ' and c_userid like ''%' || CID || '%''';
  end if;

  dbms_output.put_line(stmt);

  open ret for stmt;
  return ret;
end;
/

Later, in SQL*Plus:

set serveroutput on size 100000 format wrapped

declare
  c sys_refcursor;
  fn Customer.c_firstname%type;
  ln Customer.c_lastname %type;
  id Customer.c_userid   %type;
begin
  c := GetCustomer(LastN => 'u');

  fetch c into fn, ln, id;
  while  c%found loop
      dbms_output.put_line('First Name: ' || fn);
      dbms_output.put_line('Last Name:  ' || ln);
      dbms_output.put_line('user id:    ' || id);

      fetch c into fn, ln, id;
  end loop;

  close c;
end;
/

Edit: The comment is right, and the procedure is subject to SQL injection. So, in order to prevent that, you could go with bind variables such as in this modified procedure:

create or replace function GetCustomer(
  FirstN    varchar2 := null,
  LastN     varchar2 := null,
  CID       varchar2 := null
) return sys_refcursor
as
  stmt varchar2(4000);
  ret  sys_refcursor;

  type parameter_t is table of varchar2(50);
  parameters parameter_t := parameter_t();
begin
  stmt := 'select * from Customer where 1=1';

  if  FirstN is not null then
      parameters.extend;
      parameters(parameters.count) := '%' || FirstN || '%';
      stmt := stmt || ' and c_firstname like :' || parameters.count;
  end if;

  if  LastN is not null then
      parameters.extend;
      parameters(parameters.count) := '%' || LastN || '%';
      stmt := stmt || ' and c_lastname like :' || parameters.count;
  end if;

  if  CID is not null then
      parameters.extend;
      parameters(parameters.count) := '%' || CID || '%';
      stmt := stmt || ' and c_userid like :' || parameters.count;
  end if;


  if    parameters.count = 0 then
        open ret for stmt;
  elsif parameters.count = 1 then
        open ret for stmt using parameters(1);
  elsif parameters.count = 2 then
        open ret for stmt using parameters(1), parameters(2);
  elsif parameters.count = 3 then
        open ret for stmt using parameters(1), parameters(2), parameters(3);
  else  raise_application_error(-20800, 'Too many parameters');
  end   if;

  return ret;
end;
/

Note, that now, whatever the input, the select statement becomes something like select ... from ... where 1=1 and col1 like :1 and col2 :2 ... which is obviously much safer.

You don't necessarily need dynamic SQL just because certain where conditions don't apply when they are not present.

SELECT 
    C_FirstName, C_LastName, C_UserName, C_UserID 
FROM 
    CUSTOMER
WHERE 
    (FirstN IS NULL OR C_FirstName LIKE FirstN)
    AND (LastN IS NULL OR C_LastName LIKE LastN)
    AND (CUserName IS NULL OR C_UserName LIKE CUserName)
    AND (CID IS NULL OR C_UserID LIKE CID)

Placing this code in a stored procedure inside a package is a excellent idea.

Oracle provides some excellent documentation that can get you up to speed on stored procedures and packages. You might want to start out with the Concepts Guide to get an understanding of how Oracle works, then move on to the SQL Language Reference and PL/SQL Language Reference for information pertinent to your current task.