How do Christians tell whether they lack or have the "power" promised in Acts 1:8?

In John 15 Jesus declares that He is the true vine and that the Father is the vine dresser. Professing believers are the branches in the vine and they are there to produce fruit abundantly. The vine dresser (God) works to keep the vine healthy and increase fruit production by variously removing fruitless branches and pruning those that have some production so that it will increase.

“I am the true vine, and my Father is the vinedresser. Every branch in me that does not bear fruit he takes away, and every branch that does bear fruit he prunes, that it may bear more fruit." - John 15:1-2

Jesus then makes clear that the branch is helpless to produce fruit without maintaining vital connection to the vine, hence those who are cut off and burned have been without vital connection.

Abide in me, and I in you. As the branch cannot bear fruit by itself, unless it abides in the vine, neither can you, unless you abide in me. I am the vine; you are the branches. Whoever abides in me and I in him, he it is that bears much fruit, for apart from me you can do nothing. If anyone does not abide in me he is thrown away like a branch and withers; and the branches are gathered, thrown into the fire, and burned. - John 15:4-6

God the Father (vine dresser) is glorified when the branches bear much fruit and the bearing of much fruit by the branches demonstrates that they are disciples of Jesus.

By this my Father is glorified, that you bear much fruit and so prove to be my disciples. - John 15:8

Jesus has linked fruit bearing with abiding and now he links abiding with obedience to His commands, specifically the command to love one another as He has loved us.

If you keep my commandments, you will abide in my love, just as I have kept my Father's commandments and abide in his love. - John 15:10

“This is my commandment, that you love one another as I have loved you. Greater love has no one than this, that someone lay down his life for his friends. You are my friends if you do what I command you. No longer do I call you servants, for the servant does not know what his master is doing; but I have called you friends, for all that I have heard from my Father I have made known to you. You did not choose me, but I chose you and appointed you that you should go and bear fruit and that your fruit should abide, so that whatever you ask the Father in my name, he may give it to you.  These things I command you, so that you will love one another. -John 15:12-17

He then contrasts the hatred of the world with the love of God (vs. 18-25) and culminates the chapter with a promise that he would send them, from the Father, the Helper, the Spirit of Truth, who proceeds from the Father. This Spirit proceeds from the Father (vine dresser), bears witness about Jesus (vine), and enables disciples (fruit bearing branches) to bear witness about Jesus also.

“But when the Helper comes, whom I will send to you from the Father, the Spirit of truth, who proceeds from the Father, he will bear witness about me. And you also will bear witness, because you have been with me from the beginning. - John 15:26-27

So the Spirit proceeding from God (vine dresser) is given to the branches (disciples) through the vine (Jesus) so that much fruit (love) will be produced and remain. If the word picture may be pushed this far then the Holy Spirit is the sap that produces the love of God in the disciple through Jesus.

John further cements this in his first epistle, particularly in 1 John 4:7-21:

Beloved, let us love one another, for love is from God, and whoever loves has been born of God and knows God. Anyone who does not love does not know God, because God is love. In this the love of God was made manifest among us, that God sent his only Son into the world, so that we might live through him. In this is love, not that we have loved God but that he loved us and sent his Son to be the propitiation for our sins. Beloved, if God so loved us, we also ought to love one another. No one has ever seen God; if we love one another, God abides in us and his love is perfected in us. By this we know that we abide in him and he in us, because he has given us of his Spirit. And we have seen and testify that the Father has sent his Son to be the Savior of the world. Whoever confesses that Jesus is the Son of God, God abides in him, and he in God. So we have come to know and to believe the love that God has for us. God is love, and whoever abides in love abides in God, and God abides in him. By this is love perfected with us, so that we may have confidence for the day of judgment, because as he is so also are we in this world. There is no fear in love, but perfect love casts out fear. For fear has to do with punishment, and whoever fears has not been perfected in love. We love because he first loved us. If anyone says, “I love God,” and hates his brother, he is a liar; for he who does not love his brother whom he has seen cannot love God whom he has not seen. And this commandment we have from him: whoever loves God must also love his brother.

The witness that Christians must give to the world by the power of the Holy Spirit is love for one another. Not love as the world loves (love for friends but not enemies) but love as God in Christ has loved us: While we were yet sinners Christ died for us.

The fruit of the Spirit is love, joy, peace, forbearance, kindness, goodness, faithfulness, gentleness, self-control (Galatians 5:22-23). This singular fruit, described in it's many aspects, is not the power of the Holy Spirit itself but it is what the power of the Holy Spirit produces in the life of someone to whom that Spirit has been given. Just as a branch cannot produce fruit unless it is connected to the vine (it has no power within itself), so also no human person, no matter how pious, determined or well-meaning they may be, can produce these qualities within themselves apart from the Spirit...the power does not lie within them. This is the meaning of "Apart from me you can do nothing."

Power simply means "strength, ability". In 1 Corinthians 2 Paul says that his preaching was not with eloquent words, etc. (in other words not his power or ability) but the ability of the Spirit. "When I am weak, then I am strong." Paul did not possess within his great learning and eloquence the power to proclaim the gospel to the saving of souls. The Gospel IS the power of God for salvation (Romans 1:16) and he announced it as such.

So when Jesus tells the disciples that they would receive power when the Holy Spirit has come upon them He is telling them that they will receive the ability to do what they possess no ability within themselves to do. The criteria then, for the believer to discern whether they have received the power of the Spirit is:

1. Confession (abiding over time) that Jesus is the Son of God
2. Love for others that is not self-generated and temporary but imparted and abiding

Practices to increase this power are an ever deepening abiding in Christ and yielding to the pruning activities of the Father. We must examine ourselves daily to see whether we are in the faith. Apart from Jesus Christ, the Son of God, we can do nothing.