Well, the main question is if you know whether or not she was from the City of Aachen or from the local district around it which is also called Aachen. (This is sort of like all those people on censuses saying they are from "New York". If they mean New York City, then it means one thing... but if they mean New York State, then the search can take longer.) For example, my ancestor was from the district of Aachen, but she was from the village of Kesternich in that district and the birth record was in that village.
Anyway, first, go to
www.familysearch.org and go to the drop-down menu for "Library" at the top of the page. Select "Library Catalog". Then, select "Place Search". Then, type in for "place": "Aachen" and "part of": "Germany".
After the page loads up you will see several choices for Aachen. The first is for the city of Aachen. Some of the others are for towns in the district of Aachen (Kr. is short for "Kreis" which means district or county.) I would start with the city (first choice) and click on that. It will take you to a page which lists all the categories for the records they have on Aachen. You should probably first look in the Civil Records, so click on that. Then, the next page has the second choice for the ZIVILSTANDSREGISTER 1798-1875, AACHEN. The "Zivilstandsregister" is the city civil record of births, marriages and deaths.
After you click on that and it takes you to a page that has "Title Zivilstandsregister, 1798-1875" at the top, and "Authors Aachen, Rheinland, Burgermeisterei underneath that etc., THEN look in the upper right corner of the page for "VIEW FILM NOTES". Click on that.
"Geburten" means births. You can either order a film of the index for the year you want or, if you are pretty sure you know what year she was born, you can go directly to the "Geburten" for that year. I would recommend getting an index first if you don't know what the birth date was exactly. These records are in German and the index will make it MUCH easier to find. (You'll notice that the page runs out at Geburten 1863, but you can click on "Next film notes" at the bottom of the page to see more.) In the right hand column are the numbers for the films. For example, 938708 is the number for the microfilm index of births from 1853-1862. Once you know what films you want, go to your local LDS Family History Center and ask them to order those films for $5 each (I think) for a short-term rental. To find the closest center to your home, on that first page,
www.familysearch.org, under the same drop-down menu ("Library") select "Family History Centers" and then you can type in your town and state and it will tell you where the closest one is. I am not Mormon, but I have always found the staff at these centers to be wonderfully helpful.
PLEASE LET ME KNOW IF YOU HAVE ANY TROUBLE WITH THIS. If you find the right record, then you will probably be able to find all kinds of other family there - - siblings and ancestors. People in those days often had all their extended family from the same few villages.