It is possible, but not practical.
You could share the dial-up connection on the PC and network that to the router via Ethernet.
The problem is the extremely slow speed of a dial up connection. A dial up modem has a maximum speed of 56Kbps and the FCC limits the connection to 52Kbps. Under realistic conditions your lucky if you get a consistent 48Kbps connection.
There is some speed loss involved in transferring packets of data around on a network. A 40Kbps Internet connection is not very useful for more than very limited Internet use, no heavy graphics, streaming audio or video, etc.
For comparison, a average low end DSL plan has a 1.5Mbps connection speed which is over 30 times faster. This speed is even marginal for networking, if you want to stream smooth video from the Internet.
Inexpensive broadband DSL plans are not much more expensive than dial up. You can get DSL in many areas for under $20 a month. The DSL modem can easily be connected to a wireless router to enable WiFi broadband in your entire home.
A lot will depend on what Internet providers are available in your specific area, and what levels of service they offer in that location.
For instance DSL Internet might be better in one city, and cable Internet options better in another. DSL is generally cheaper than cable Internet, however the cable plans are usually faster.
If you are interested in broadband Internet then go to a ISP comparison site that lets you input your address, http://isp1.us/find/ and compare the pricing and terms of the offers that are available in you area.
You'll want to pick the fastest plan that fits your budget. In some areas this can be 100Mbps+. But these high end broadband plans are expensive. You may find that something in the 1.5 to 20Mbps range might satisfy your needs at a much lower cost.
Good Luck