For example, if your wife would prefer not to use a public one and wants to install one in your house? Is there anything to be aware of if you're considering doing this?
The question seems to be not if it it theoretically possible to build a private mikva, but practically how to go about doing so. This is beyond the scope of a Q&A website, and that is not a new thing. The Kitzar Shulchan Aruch writes about this:
דִּינֵי הַמִּקְוֶה, רַבִּים הֵם מְאֹד. וּבְכָל מָקוֹם שֶׁעוֹשִׂין מִקְּוֶה, אֵין לַעֲשׂוֹתוֹ כִּי אִם עַל יְדֵי רַב מֻמְחֶה לָרַבִּים, גָּדוֹל בַּתּוֹרָה וּבְיִרְאָה. וְכַאֲשֶׁר יִתְהַוֶּה בוֹ אֵיזֶה שִׁנּוּי גָּדוֹל אוֹ קָטָן, יַעֲשׂוּ מִיָד שְׁאֵלַת חָכָם. וְכֵן כַּאֲשֶׁר יִצְטָרְכוּ לִשְׁאֹב אוֹתוֹ לְנַקּוֹתוֹ, יִשְׁאֲלוּ אֵיךְ יִתְנַהֲגוּ.
The laws regarding mikvah are very numerous. Wherever a mikvah is being made, it should be constructed only under the supervision of a very renowned poseik, great in Torah and in fear of God. When any change occurs (in the mikvah), whether a minor or major (change), a competent poseik should be consulted immediately. Also, when it becomes necessary to draw the water in order to clean it, a poseik should be consulted about the proper way to do it.
Maybe this will help, it gives the basics of what the Mikvah must have. I know Asher Meza had a video on someone building their own mikvah https://www.youtube.com/watch?v=JaLfc4VZIAM
If it was built with the consultation and guidance of a knowledgeable Posek. There is another issue, that one would need a Mikava attendant to come in to make sure that the woman has toivled K'Halacha. The Noda BiYehuda permits a husband to do this and even to help her go to Mikva when there is no one else around. However, if a woman is perfectly able to go to a regular Mikva or call an attendant from elsewhere for her private Mikva, her husband CANNOT be used just because she feels uncomfortable going to Mikva in front of her. Please ask your local Rabbi as every case could have additional factors which we are unaware of.
it is not so hard for a mikva to become pasul - invalid - and without the constant supervision of a competent authority - which public mikvahs have - one can unknowingly transgress countless avairos each punishable by kores. who would want to take such a risk. modern mikvahs are more hygienic clean and comfortable than anything a private person would venture on his own.