How can I view IPv6 router advertisements that are being received by my computer for diagnostic purposes?

Using tcpdump which is installed by default on many distributions:

tcpdump -n -i eth0 icmp6

will show you all ICMPv6 packets of which - under usual conditions - almost all are neighbor discovery packets. In order to see only router advertisements, use the following command:

tcpdump -n -i eth0 icmp6 and ip6[40] == 134

For more verbosity, add -v; to display packet contents, use the option -X.

tshark is usually bundled with wireshark, which most distributions do not install by default but provide as additional package.

While not built in, I find the command-line tool radvdump (part of the radvd package on my distro) very informative as it fully decodes the router advertisements and displays them in detail, including advertised DNS information.

$ radvdump

interface eth0
{       
    AdvSendAdvert on;
    # Note: {Min,Max}RtrAdvInterval cannot be obtained with radvdump
    AdvManagedFlag on;
    AdvOtherConfigFlag on;
    AdvReachableTime 0;
    AdvRetransTimer 0;
    AdvCurHopLimit 64;
    AdvDefaultLifetime 65535;
    AdvHomeAgentFlag off;
    AdvDefaultPreference medium;
    AdvSourceLLAddress on;
    AdvLinkMTU 1280;
    AdvIntervalOpt on;

    prefix 1111:2222:3333:4444::/64
    {       
        AdvValidLifetime infinity; # (0xffffffff)
        AdvPreferredLifetime infinity; # (0xffffffff)
        AdvOnLink on;
        ...

The most common ones are tcpdump, wireshark and tshark (the command-line version of Wireshark). Those tools can capture and decode network traffic, including Router Advertisements.