Hi Rajat,

I have gone through your problem.Please try the below code.
 

VF Page:

<apex:page standardController="Account" recordSetVar="Accountvar" action="{!ViewData}" extensions="AccountListViewController">

<apex:sectionHeader title="My Accounts" subtitle="Account List View"/>
    <apex:form >
        <apex:pageBlock >
          <apex:pageMessages id="error" />

           <apex:panelGrid columns="7" id="buttons" >
           <!---<apex:pageBlockButtons>---->
                <apex:commandButton reRender="error,blocktable,buttons" action="{!Save}" value="Save"/>
                <apex:commandButton reRender="error,blocktable,buttons" action="{!Cancel}" value="Cancel"/>
                <apex:inputHidden />
                <apex:commandButton reRender="error,blocktable,buttons" disabled="{!!hasprevious}" action="{!First}" value="First"/>
                <apex:commandButton reRender="error,blocktable,buttons" disabled="{!!hasprevious}" action="{!Previous}" value="Previous"/>
                <apex:commandButton reRender="error,blocktable,buttons" disabled="{!!hasnext}" action="{!Next}" value="Next"/>
                <apex:commandButton reRender="error,blocktable,buttons" disabled="{!!hasnext}" action="{!Last}" value="Last"/>
           <!---</apex:pageBlockButtons>--->
           </apex:panelGrid>

           <apex:pageBlockSection id="blocktable" >

                <apex:pageBlockTable value="{!AccountsortList}" var="t" rendered="{!NOT(ISNULL(AccountsortList))}" id="cmdsort">

                        <apex:column >
                            <apex:facet name="header">   
                                <apex:commandLink action="{!ViewData}" value="Account Name{!IF(ExpressionSort=='name',IF(DirectionOfSort == 'ASC', '▼', '▲'),'')}">
                                    <apex:param value="name" name="column" assignTo="{!ExpressionSort}" ></apex:param>
                                </apex:commandLink>
                            </apex:facet>
                            <apex:outputLink value="/{!t.Id}" target="_blank">{!t.Name}</apex:outputLink>
                        </apex:column>


                        <apex:column headerValue="BillingState/Province" value="{!t.BillingState}"/> 
                        <apex:column headerValue="Phone" value="{!t.Phone}"/>
                        <apex:column headerValue="Type" value="{!t.Type}"/>                   
                        <apex:column headerValue="Account Owner Alias" value="{!t.Owner.Name}"/>
                        <apex:column headerValue="Website" value="{!t.Website}"/>

                    <apex:inlineEditSupport event="onClick"/>


                </apex:pageBlockTable>

           </apex:pageBlockSection>   


        </apex:pageBlock>
    </apex:form>
</apex:page>

Apex Class:AccountListViewController

public class AccountListViewController{
public List<Account> AccountsortList {get; set;}
public String SortingExpression = 'name';
public String DirectionOfSort = 'ASC';

    public AccountListViewController(ApexPages.StandardSetController controller) {
        AccountsortList = new List<Account>();
        ApexPages.StandardSetController ssc = new ApexPages.StandardSetController(AccountsortList);
    }

    public String ExpressionSort {
        get {
            return SortingExpression;
        }
        set {
            If(value == SortingExpression) {
                DirectionOfSort = (DirectionOfSort == 'ASC')? 'DESC' : 'ASC';
            }
            else {
                DirectionOfSort = 'ASC';
                SortingExpression = value;
            }
        }

    }

    public String getDirectionOfSort() {
        If(SortingExpression == Null || SortingExpression == '') {
            return 'DESC';
        }
        else {
            return DirectionOfSort;
        }
    }

    public void setDirectionOfSort(String value) {
        DirectionOfSort = value;
    }

    public List<Account>getAccounts() {
        return AccountsortList;
    }

     public PageReference ViewData() {
        String FullSortExpression = SortingExpression + ' ' + DirectionOfSort;
        system.debug('SortingExpression:::::'+SortingExpression);
        system.debug(DirectionOfSort);

       String Queryitem = ' SELECT Id, Name, Phone, BillingCountry,BillingState, Type, Owner.Name, Website FROM Account WHERE Account.Name != Null ORDER BY ' + FullSortExpression +' Limit 1000';
       system.debug(Queryitem);

        AccountsortList = DataBase.query(Queryitem);
        system.debug(AccountsortList);
        return Null;
    }
}

I hope you find the above solution helpful. If it does, please mark as Best Answer to help others too.

Thanks and Regards,
Sachin Arora