There are two ways of looking at the subject, and both are legitimate ways to look at it.
To choose a wife, you look at what scripture teaches, and make a decision.
To choose a wife, you pray and seek God until you know His will about this specific situation, and then make a decision.
For me, it was a combination of the two.
Proverbs is written to a young man. That is important to keep in mind when looking for a wife. It warns against the adulterous, the quarrelsome woman, and the foolish woman, and speaks well of the wise woman, the woman who fears the Lord, and the noble woman. There is a whole chapter about the type of woman that is desirable as a wife, and it's easy to find right there at the end of the book. Women like this chapter, but it is a great chapter for men looking for wives and for husbands, too. It was written to a young man after all.
There are other criteria she should have. She needs to be a believer. You shouldn't marry someone whom "God hath joined together" with someone else. Marrying a virgin is desirable. If a man has taken a woman's virginity, there is a moral issue there to consider as well. In the Old Testament, such a man was required to marry the girl if her father permitted it. In the Bible, virginity is a desirable trait.
The Bible says "...if you marry you have not sinned...." As long as you don't sin somehow in who you choose or how you do it, it's not a sin to marry. You could sin by marrying a divorced woman who is supposed to go back to her husband. You could sin by marrying a close relative. You could sin by marrying in an dishonorable way that does not honor either of your parents. But if you do things right, marry a believer who is eligible in a right way, it's not a sin. You don't have to hear a voice from heaven.
Then there is getting direction from God. Abraham prayed that the Lord would send His angel before his servant to help him find a wife for Isaac. The servant prayed for a very specific set of actions for the woman the Lord had chosen for Isaac to perform, offering to give drink to him and to his camels. This was an incredibly tiresome task, they say, which could require pulling dozens of buckets up from the well since camels would be very thirsty after a trip across the desert (and I am not sure if that was the route taken, since armies came from the north, since they followed the rivers along, but I still assume it was a lot of work.) This also showed that she was generous and took care of people. It might have shown a bit about her hospitality, which is also a sign of good character. Rebecca was also a virgin. She was beautiful as well.
When God answered Abraham's servant's prayer and he knew she was from his clan, he proposed marriage on behalf of Isaac.
There are numerous examples of God giving people specific instructions. The Spirit spoke to certain people in the New Testament, and as believers we are given some rather amazing promises of God answering when we ask in faith. So you can pray and seek the Lord for specific direction on who to marry.
"All things are possible to him that believes."
At some point, though, you have to make a decision. It doesn't make sense to me to reason, "We are going out now. It must be God's will for us to be together" and then two months later think, "We are not going out now. It must not be God's will for us to be together."
If something is difficult, that doesn't mean it is not God's will either.
In my case, my wife had characteristics I was looking for, even some criteria I found in scripture that seemed hard to find, and fit with a lot of my personal tastes. She was pretty, hard-working, a virgin, feared the Lord, and various other things I had prayed for, and a lot of things I hadn't thought to pray for. I think I met more characteristics of things she'd pray for.
And it actually seemed like the Lord was telling me to marry her when I prayed. I'd forgotten about it, but I'd recalled, when I wrote in my diary back then, that it seemed like the Lord was speaking to me that I would meet the woman I would marry that month, and that she would be 23 years old. She was indeed 23. She got upset about something on the phone once, and I prayed and it seemed like the Lord was saying she was upset because such and such happened to her. Later I talked with her, and that thing had happened to her. She said the Lord was speaking to her out of a certain verse about us about the threefold strand. Later, other people kept telling us about that verse, even someone in the US who prayed for us at a party for our wedding after we'd gotten married. It seemed like the Lord was telling me yes when I asked if I should marry her, and eventually 'Yes, why do you not believe Me?" (Ouch.)
I was nearly sure, maybe 95%, but I had that remaining bit of doubt. Eventually, I took a step of faith and made a decision. I prayed and gave the Lord my reasons for proposing and told Him I was going to do it, and if He didn't want me to propose, to stop me or to tell me to stop. After that, I was completely sure. It was a settled matter.
I was hoping for some prophetic confirmation before I made my decision. The next day, someone did prophesy about the Lord doing a work in us that would last our whole lives and that we would go to many places and minister to many people. Parts of that have been fulfilled.
Anyway, a good theme verse is this one,
Proverbs 3
5 Trust in the Lord with all your heart
and lean not on your own understanding;
6 in all your ways submit to him,
and he will make your paths straight.
(NIV)
Something important is to submit to the Lord, or acknowledge Him as another translation says, in all your ways. That means praying about... not just rushing into something without praying about because you don't have peace and are afraid God will be against it. But really pray about it. Pray for wisdom. The great thing about praying for wisdom is that we are promised it if we pray to God in faith without doubting in James 1.
If you do these things the vers says, He will make your paths straight. The KJV says that He will direct thy paths. That's a great promise.
Do all these things, and you can make a decision and trust God to help you make the right one. Remember, no decision is a decision, too. So not making a decision isn't the way out of decision-making.
So when you pray about something like this, study scriptures to learn principles, be sensitive to what the Lord is speaking to your heart, to godly council, to how you think He may be directing you through circumstances and answering your prayers. Then tell God what decision you are making and why, and ask Him to stop or not let you have peace about it if you are choosing wrongly. Then go forward.
If it's marriage, you should probably get both sets of parents involved before going too far with making the decision. If you have godly parents its good to get their input before popping the question. You can ask other people you trust, especially people with life experience who have been married, if they think you are a good match and if she is a good girl for you and if you are a good man for her.