There are two ways that I can come up with to answer this question:-
1st way to know that you have received the holy spirit(i.e reborn of spirit) is by trusting what the bible says. When you accept the bible to be the word of God and repent for your sins and believe that you are saved by believing in Jesus, you also must have faith in what the bible promises. there are verses where Jesus promises the gift of the Holy Sprit.
Jesus says :
John 14:15-16 (NIV)
If you love me, keep my commands. And I will ask the Father, and he will give you another advocate to help you and be with you forever— the Spirit of truth.
John 16:7 (NIV)
But very truly I tell you, it is for your good that I am going away. Unless I go away, the Advocate will not come to you; but if I go, I will send him to you.
This is a promise to us that on believing (i.e loving him and following his commands) "in the name of Jesus" (explicitly mentioned in John 14:26) we will receive the Holy Spirit for sure.
2nd way actually makes you believe in the promise that Jesus gives, by your experiences. As also rightly put by olskeevi, Holy Spirit lives in you and communicates through your heart (my belief). Holy Spirit does change you and the change can be in many aspects, may be physical, your priorities may change, your longings for things may change, your attitude/behavior may change etc. But there will be change which must be noticeable atleast to you! (coz many changes in my life were evident to me and not to people around me, coz they don't know everything about me.)
Lastly when I read
1 John 2:5-6
But if anyone obeys his word, love for God is truly made complete in them. This is how we know we are in him: Whoever claims to live in him must live as Jesus did.
I come to know that, to know whether I am a Christian (meaning I have the Holy Spirit living in me) or not, I will have to do what Jesus did (i.e, live like him by doing God's will). By doing god's will I will come closer to him an love him even more. Also 1 thing that we must remember is that "the will of the God also must be in accordance to the teachings of the bible!"