You cannot get the entire thing only with a standard controller, there are multiple steps in this process.

Step 1 - Create a new class that will serve as an extension to Lead. The class should have two things - (A) a "searchText" property, that we use for the search text, and (B) a "search" method, that does the actual search. Here's what I could get :-

public class SearchController
{
private apexpages.standardController controller {get; set; }
private Lead l;
public List<Lead> searchResults {get; set; }
public string searchText
{
get
{
if (searchText==null) searchText = '';
return searchText;
}
set;
}

public SearchController(ApexPages.StandardController controller)
{
this.controller = controller;
this.l = (Lead) controller.getRecord();
}

public PageReference search()
{
if(SearchResults == null)
{
SearchResults = new List<Lead>();
}
else
{
SearchResults.Clear();
}

String qry = 'Select Id, Name from Lead where name like \'%'+searchText+'%\') Order By Name';
SearchResults = Database.query(qry);
return null;
}
}

Step 2:- Add he controller to your page as an extension

<apex:page standardController="Lead" extensions="SearchController">

 Step 3:- Hook up the searchText and the output grid

    <apex:form id="searchForm">
        <apex:PageBlock mode="edit">        
        <apex:pageblockSection id="searchBlockSection">
            <apex:pageBlockSectionItem id="searchBlockSectionItem">
                <apex:outputLabel >Keyword</apex:outputLabel>
                <apex:panelGroup >
                    <apex:inputtext id="searchTextBox" value="{!searchText}">
                        
                    </apex:inputtext>
                    <apex:commandButton Id="btnSearch" action="{!search}" rerender="renderBlock" status="status" title="Search" value="Search">                    </apex:commandButton>
                </apex:panelGroup>
            </apex:pageBlockSectionItem>
        </apex:pageblockSection>
        <apex:actionStatus id="status" startText="Searching... please wait..."/>        
        <apex:pageBlocksection id="renderBlock" >
            <apex:pageblocktable value="{!SearchResults}" var="o" rendered="{!NOT(ISNULL(SearchResults))}">
                <apex:outputLink value="/{!o.Id}">{!o.Name}</apex:outputLink>
                <apex:column value="{!o.Id}"/>
            </apex:pageblocktable>        </apex:pageBlocksection>

This should give you a search box. I haven't tested the code, so you might have to fix some syntax errors etc., but this is the general direction you can take to solve this problem.